a precision measurement of the weak mixing angle in møller ... · apv = −158 ppb±21 ppb (stat)...
TRANSCRIPT
A Precision Measurement of the Weak Mixing Angle inMøller Scattering at Low Q2
Thesis by
G. Mark Jones
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
2004
(Defended May 19, 2004)
ii
c© 2004
G. Mark Jones
All Rights Reserved
iii
Acknowledgements
I am greatly indebted to many people that have provided support, encouragement, insight,
and guidance to me over the course of the work presented in this thesis. I would like first to
thank my advisor, Emlyn Hughes, for allowing me the opportunity to be a part of his group
at Caltech. I was afforded the chance to receive an excellent education in the classroom
and in the laboratory, and for that I am truly grateful.
The success of the challenging E158 experiment is due to the many diverse contributions
of the collaborators. To each of them I owe a great deal of thanks. In particular, I would
like to acknowledge Krishna Kumar, Paul Souder and my advisor for their able leadership
at SLAC. Their immense physical insight and experience were crucial to the experiment. I
am also grateful for the tireless efforts of Yury Kolomensky and Michael Woods. Without
their keen ability, the experiment could not have succeeded. Also, both Yury and Mike
were patient teachers, a trait greatly appreciated by the students on the experiment. I am
thankful that I got to know Raymond Arnold, who reminded me in the midst of all the
chaos of the experiment that I shouldn’t forget that the study of physics is an interesting
and worthwhile endeavor. Also, I would like to acknowledge Michael Olson ”MO” for being
an excellent run coordinator and teaching me the importance of the National Football
League and the Packers. I am thankful to Zenon Szalata for all the assistance he gave me,
especially on those owl shifts where I would inevitiably have to call him at his home for
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advice. I would also like to acknowledge John Weisend who did a superb job with the liquid
hydrogen target, and who also taught me how to remain calm in the midst of warning bells
and flashing lights. I would like to thank Dieter Walz, whose unbounded knowledge about
the workings of the laboratory and his interesting tales made many a shift go by quickly.
The construction of the apparatus in End Station A within the time constraints of the
experiment proved to be a truly formidable undertaking. I am grateful for the diligent
work of Ken Baird that saw the experiment go from an idea on paper to existing in the
real world. I would also like to thank Michael Racine and Carl Hudspeth, as well as the
rest of the craftsmen who built the experiment, whose skill and expertise in all manners of
construction allowed the apparatus to function as designed.
I owe many thanks to my fellow students and the post-docs, who worked with me on
the data analysis, and also helped in taking all those shifts. In particular, I want to thank
Carlos Arroyo, Klejda Bega, Mark Cooke, Waled Emam, Brian Humensky, Lisa Kaufman,
Peter Mastromarino, Kent Paschke, Jaideep Singh, Baris Tonguc, Brock Tweedie, Imran
Younus, and Antonin Vacheret. We went through a lot together, and I feel privileged to
have worked with each of them.
I am truly grateful for having been able to work with Clive Field on the luminosity
monitor. I was always in awe of the combination of his sharp mind and his immense
technical ability. He taught me how to systematically approach problems, and he showed
me new ways of thinking about things. I would also like to acknowledge the contribution
of Gholamali Mazaheri, who worked diligently on the electronics side of the detector.
I was fortunate to also work with many interesting people during my time at Caltech.
David Pripstein introduced me to the work in the lab, and he also helped me adjust to life
in graduate school in general. His outgoing personality and his taste for fine wines made
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my first year at Caltech memorable and enjoyable. It is no exaggeration to say that my
interactions with Dave as an undergraduate led me to choose Caltech for graduate school.
For that, I am very thankful. Steffen Jensen and Jaideep Singh were always able to make
working in the lab more enjoyable with their sense of humor. I want to thank Tina Pavlin
for being a good listener and making sure I didn’t focus too much on lab work. Also, many
thanks to Georgia Frueh for skillfully managing all of the graduate students in our group
with a smile.
My experiences at the University of Kentucky as an undergraduate greatly influenced
my choice to pursue the study of physics. In particular, Wolfgang Korsch was a very
positive influence on me. His love of physics, as well as his patient guidance, gave me
great encouragement. The lessons he gave me in the classes he taught and at Jefferson lab
proved to be invaluable in graduate school. I would also like to mention Valdis Zeps and
Tim Gorringe, two excellent physics professors I had as an undergraduate. Their skillful
instruction prepared me for the rigors of graduate school and instilled in me a desire to
learn more. I would also like to thank Jian-ping Chen at Jefferson lab, who made my time
there both enjoyable and enlightening.
I owe special thanks to my friends and family. Peter Mastromarino, with his sense of
humor and genuine good nature, proved to be a true friend throughout graduate school. I
feel very fortunate that we ended up on the same experiment together. I am thankful to
Ted “Theodore” Corcovilos for his friendship, as well as helping me survive classwork at
Caltech and teaching me to “chill.” Sean Stave and Aaron Skaggs also provided me with
much needed encouragement, probably more than they know. And finally, I would like to
give thanks to my family, Gary, Linda, and Chris, to whom I literally owe everything. They
have always given me unconditional support and encouragement. I hope they feel they are a
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part of the work in this thesis, because I know that I could not have accomplished anything
without them.
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Abstract
The electroweak theory has been probed to a high level of precision at the mass scale of
the Z0 through the joint contributions of LEP at CERN and the SLC at SLAC. The E158
experiment at SLAC complements these results by measuring the weak mixing angle at a
Q2 of 0.026 (GeV/c)2, far below the weak scale.
The experiment utilizes a 48 GeV longitudinally polarized electron beam on unpolarized
atomic electrons in a target of liquid hydrogen to measure the parity-violating asymmetry
APV in Møller scattering.
The tree-level prediction for APV is proportional to 1− 4 sin2 θW . Since sin2 θW ≈ 0.25,
the effect of radiative corrections is enhanced, allowing the E158 experiment to probe for
physics effects beyond the Standard Model at the TeV scale.
This work presents the results from the first two physics runs of the experiment, covering
data collected in the year 2002. The parity-violating asymmetry APV was measured to be
APV = −158 ppb± 21 ppb (stat) ± 17 ppb (sys). (1)
The result represents the first demonstration of parity violation in Møller scattering. The
observed value of APV corresponds to a measurement of the weak mixing angle of
sin2 θeffW = 0.2380 ± 0.0016 (stat) ± 0.0013 (sys), (2)
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which is in good agreement with the theoretical prediction of
sin2 θeffW = 0.2385 ± 0.0006 (theory). (3)
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Contents
Acknowledgements iii
Abstract vii
1 Introduction 1
1.1 Early Study of the Weak Force . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Parity Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Electroweak Unification . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 LEP and SLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 The Role of the E158 Experiment . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 E158 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Experiment Timeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Theory 8
2.1 The E158 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Previous Low Q2 Electroweak Measurements . . . . . . . . . . . . . . . . . 9
2.2.1 NuTeV Experiment Overview . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Atomic Parity Violation Overview . . . . . . . . . . . . . . . . . . . 11
2.3 E158 Experiment at Tree Level . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Radiative Corrections to APV . . . . . . . . . . . . . . . . . . . . . . . . . . 14
x
2.4.1 γ − Z0 Mixing Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.2 Heavy Box Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.3 γ − Z0 Box Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.4 ρ Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.5 Definition of sin2θW (Q2) . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 E158 Beamline and Beam Monitoring 20
3.1 Polarized Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.1 Laser Bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2 Combiner Bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.3 Wall Bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.4 Polarized Gun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.5 Helicity Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.6 Beam Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Beam Position Monitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 BPM Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.2 BPM Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Charge Monitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1 Toroid Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.2 Toroid Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Wire Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Skew Quadrupole Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 Liquid Hydrogen Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7 Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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3.7.1 Dipole Chicane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.7.2 Main Acceptance Collimator . . . . . . . . . . . . . . . . . . . . . . 44
3.7.3 Quadrupole Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7.4 Insertable Acceptance Collimator . . . . . . . . . . . . . . . . . . . . 46
3.7.5 eP Collimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.7.6 Synchrotron Collimation and Photon Masks . . . . . . . . . . . . . . 49
4 Detectors 52
4.1 E158 Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.1 Calorimeter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.2 Calorimeter Electronics . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.3 Calorimeter Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.4 Calorimeter Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Polarimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 Polarimetry Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.2 Polarimeter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.3 Polarimeter Systematic Uncertainties . . . . . . . . . . . . . . . . . . 64
4.2.4 Polarimeter Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Profile Monitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.1 Profile Monitor Design . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.2 Profile Scan Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Pion Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4.1 Pion Detector Design . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4.2 Pion Detector Resolution . . . . . . . . . . . . . . . . . . . . . . . . 70
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4.4.3 Pion Detector Electromagnetic Background . . . . . . . . . . . . . . 71
4.4.4 Ratio of Nπ to Nee . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4.5 Pion Detector Results . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5 Synchrotron Light Monitor . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5.1 Synchrotron Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5.2 SLM Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5.3 SLM Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.5.4 SLM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Luminosity Monitor 80
5.1 Detector Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Synchrotron Radiation Background . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Detector Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4 Lumi Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5 Gas System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.6 Asymmetry Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.7 Resolution Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.8 Target Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.9 Synchrotron Radiation Suppression . . . . . . . . . . . . . . . . . . . . . . . 97
5.10 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.10.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.10.2 Measured Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.11 Missing Pulse Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.12 Lumi as a BPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
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6 Asymmetry Analysis 107
6.1 Initial Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Detector Channel Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4 Beam Dithering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.5 Data Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.6 Møller Detector Asymmetry Analysis . . . . . . . . . . . . . . . . . . . . . . 116
6.6.1 Systematic Reversals . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.6.2 Beam Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.7 Asymmetry Corrections and Uncertainties . . . . . . . . . . . . . . . . . . . 122
6.7.1 The Electron-Proton Scattering Correction . . . . . . . . . . . . . . 122
6.7.2 Beam Asymmetry Correction Systematic Uncertainties . . . . . . . . 126
6.7.2.1 First-Order Beam Correction Systematic Uncertainties . . 126
6.7.2.2 Higher-Order Beam Correction Systematic Uncertainties . 129
6.7.3 Dipole Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.7.4 Spot Size Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.7.5 Pion Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.7.6 Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.7.7 Neutral Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.7.8 Scale Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.8 Determination of APV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.9 Luminosity Monitor Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.9.1 Lumi Beam Correction Systematic Uncertainty . . . . . . . . . . . . 143
6.9.2 Lumi Dipole Contamination . . . . . . . . . . . . . . . . . . . . . . . 147
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6.9.3 Lumi Dilutions and Scale Factors . . . . . . . . . . . . . . . . . . . . 148
6.10 Lumi Asymmetry Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7 The Weak Mixing Angle 150
7.1 Extraction of sin2θW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.2 Physics beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . . 154
7.2.1 Electron Compositeness Limit . . . . . . . . . . . . . . . . . . . . . . 154
7.2.2 Scalar Doubly Charged Higgs Limit . . . . . . . . . . . . . . . . . . 156
7.2.3 Extra Z Boson Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2.4 Oblique Parameter X Limit . . . . . . . . . . . . . . . . . . . . . . . 159
7.3 Future Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.3.1 E158 Run III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.3.2 QWeak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.3.3 DIS-Parity at JLab . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.3.4 The Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . 162
7.3.5 Next Linear Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Bibliography 164
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List of Figures
1.1 Fermi’s four-point interaction for β decay. . . . . . . . . . . . . . . . . . . . . 2
1.2 The W− incorporated into β decay. . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 The running of sin2θW with four-momentum transfer Q. . . . . . . . . . . . . 6
1.4 General diagram of the E158 apparatus. The regions of the E158 calorimeter
are named as follows: A.) In ring B.) Mid ring C.) Out ring and D.) eP detector. 6
2.1 The running of sin2θW with Q. . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Neutrino scattering diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Anti-neutrino scattering diagrams. . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Tree-level contributions to the nuclear potential. . . . . . . . . . . . . . . . . 11
2.5 Tree-level contributions to APV . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Photon-Z0 mixing diagrams and the W contribution to the anapole moment. 15
2.7 Heavy boson box diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.8 Photon-Z0 box diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.9 Anapole moment contribution from the Z0. . . . . . . . . . . . . . . . . . . . 17
3.1 Location of the polarized source room. . . . . . . . . . . . . . . . . . . . . . . 20
3.2 E158 optics configuration at the source. . . . . . . . . . . . . . . . . . . . . . 21
3.3 Helicity of the E158 electron beam. . . . . . . . . . . . . . . . . . . . . . . . 24
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3.4 Integrated charge asymmetry measured near the source, spanning all of Run
1 and Run 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Run 1 and Run 2 target region integrated charge asymmetry. . . . . . . . . . 26
3.6 Location of E158 beam monitors. . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.7 E158 beam position monitor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.8 BPM signals after mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.9 Angle and position BPM resolutions. . . . . . . . . . . . . . . . . . . . . . . . 30
3.10 Energy BPM resolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.11 Toroid resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.12 Toroid signal versus calibrator signal. . . . . . . . . . . . . . . . . . . . . . . 35
3.13 Wire array profiles for both X and Y axes. . . . . . . . . . . . . . . . . . . . 36
3.14 Detector spot size correlation coefficients. . . . . . . . . . . . . . . . . . . . . 37
3.15 Location of skew quadrupole magnet. . . . . . . . . . . . . . . . . . . . . . . 39
3.16 Effect of the skew quad on detector resolution. . . . . . . . . . . . . . . . . . 39
3.17 E158 cryotarget loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.18 Cryotarget scattering chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.19 E158 spectrometer overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.20 Regions of the E158 calorimeter. . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.21 E158 dipole chicane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.22 First photon collimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.23 Main acceptance collimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.24 Effect of main acceptance collimator on signal flux. . . . . . . . . . . . . . . . 46
3.25 Profile scans with and without quadrupole magnets. . . . . . . . . . . . . . . 46
3.26 Insertable acceptance collimator. . . . . . . . . . . . . . . . . . . . . . . . . . 47
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3.27 Profile scans with (left) and without (right) the insertable collimator. . . . . 48
3.28 Insertable eP collimator, top view. . . . . . . . . . . . . . . . . . . . . . . . . 49
3.29 The hatched area represents the coverage of the eP collimator on the face of
the E158 calorimeter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.30 Profile scans without (left) and with (right) the eP collimator in place. . . . 50
3.31 Drift pipe synchrotron and photon collimators. . . . . . . . . . . . . . . . . . 51
4.1 Overhead view of the detector locations in End Station A. . . . . . . . . . . 52
4.2 Overhead view of the movable detector cart. . . . . . . . . . . . . . . . . . . 53
4.3 Partially constructed E158 calorimeter. . . . . . . . . . . . . . . . . . . . . . 54
4.4 Calorimeter channel map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5 Light guide configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.6 E158 calorimeter lead shielding. . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.7 E158 calorimeter electronics diagram. . . . . . . . . . . . . . . . . . . . . . . 57
4.8 Møller detector asymmetry resolution. . . . . . . . . . . . . . . . . . . . . . . 58
4.9 Møller plus Out ring asymmetry resolution. . . . . . . . . . . . . . . . . . . . 58
4.10 eP ring asymmetry resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.11 Polarimeter asymmetry distribution for a single run. . . . . . . . . . . . . . . 62
4.12 Polarimeter position on the detector cart. . . . . . . . . . . . . . . . . . . . . 63
4.13 Quartz-tungsten sandwich of polarimeter. . . . . . . . . . . . . . . . . . . . . 64
4.14 Run II beam polarization measurements, spanning 1 month. . . . . . . . . . 66
4.15 Profile monitor schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.16 Quartz scanner schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.17 Radial scan produced with the profile monitor. . . . . . . . . . . . . . . . . . 68
4.18 Pion detector layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
xviii
4.19 Pion detector asymmetry distribution, covering one run. . . . . . . . . . . . . 71
4.20 SLM location in the A-Line bend. . . . . . . . . . . . . . . . . . . . . . . . . 74
4.21 SLM layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.22 Regressed SLM channel 2 asymmetry distribution, for one run. . . . . . . . . 77
4.23 SLM asymmetry results for Run I and Run II. . . . . . . . . . . . . . . . . . 78
5.1 Layout of End Station A for experiment E158. . . . . . . . . . . . . . . . . . 81
5.2 Components of lumi signal at face of detector. . . . . . . . . . . . . . . . . . 81
5.3 Contributions to lumi asymmetry. . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 GEANT simulation of photon flux at the lumi. . . . . . . . . . . . . . . . . . 84
5.5 A. Front view of one full lumi ring, with sensitivity between 7 and 10 cm. B.
Side view, depicting the two lumi rings and the aluminum showering material. 85
5.6 Individual chamber design, with signal plates shaded. . . . . . . . . . . . . . 85
5.7 Lumi electronics setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.8 Front lumi signal traces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.9 Back lumi signal traces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.10 Gas system configuration for lumi. . . . . . . . . . . . . . . . . . . . . . . . . 89
5.11 X and Y correlation coefficients. The averages are shown as straight lines. . . 90
5.12 dX and dY correlation coefficients The averages are shown as straight lines. . 91
5.13 Left: Raw lumi asymmetry distribution. Right: Regression-corrected lumi
asymmetry distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.14 Extracted boiling noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.15 Synchrotron radiation background levels by chamber. . . . . . . . . . . . . . 98
5.16 Lumi signal versus beam current. . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.17 Full range of results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
xix
5.18 Tight beam cuts subset of linearity results. . . . . . . . . . . . . . . . . . . . 103
5.19 Charge normalized lumi signal following a missing pulse. . . . . . . . . . . . 104
5.20 The lumi asymmetry distribution without (left) and with (right) a cut after a
missing pulse. RMS is the statistical width of the distribution, while Sigma
refers to the width determined from the fit. . . . . . . . . . . . . . . . . . . . 104
5.21 Calculated beam position using the lumi, versus position position at lumi
figured using angle and position bpms. . . . . . . . . . . . . . . . . . . . . . . 106
6.1 Møller detector asymmetry versus Y position asymmetry. . . . . . . . . . . . 109
6.2 Møller detector asymmetry versus Y position asymmetry after regression. . . 111
6.3 Møller detector resolution with and without regression. . . . . . . . . . . . . 111
6.4 Location of components used for beam dithering. . . . . . . . . . . . . . . . . 112
6.5 A lumi chamber responding to a position dither cycle. . . . . . . . . . . . . . 113
6.6 Run I Møller detector asymmetry versus slug. . . . . . . . . . . . . . . . . . . 117
6.7 Run II Møller detector asymmetry versus slug. . . . . . . . . . . . . . . . . . 118
6.8 Measured asymmetry for each energy-halfwave plate setting. . . . . . . . . . 119
6.9 Run I Møller detector asymmetry versus slug, sign flips suppressed. . . . . . 120
6.10 Run II Møller detector asymmetry versus slug, sign flips suppressed. . . . . . 121
6.11 Data and simulation comparison with normal running conditions. . . . . . . 123
6.12 Data and simulation comparison with the insertable acceptance collimator in
position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.13 Profile scan with the Møller and electron-proton scattering simulation results
superimposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.14 Asymmetry result from the eP detector in Run I. . . . . . . . . . . . . . . . . 126
6.15 Run I Out ring asymmetry data. . . . . . . . . . . . . . . . . . . . . . . . . . 130
xx
6.16 Run II Out ring asymmetry data. . . . . . . . . . . . . . . . . . . . . . . . . 130
6.17 Run I asymmetry plotted versus channel number for 46 GeV running. . . . . 135
6.18 Run I asymmetry plotted versus channel number for production data. . . . . 136
6.19 Run I lumi asymmetry data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.20 Run II lumi asymmetry data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.21 Run I and Run II lumi asymmetry with systematic reversal sign flips ignored. 144
6.22 Run I lumi asymmetry plotted versus channel number. . . . . . . . . . . . . . 147
7.1 Bremsstrahlung diagrams included in Fb(y). The crossed versions must also
be computed, for a total of 16 diagrams. . . . . . . . . . . . . . . . . . . . . . 151
7.2 Experimental results and the theoretical running of the weak mixing angle. . 153
7.3 The measured weak mixing angle evolved to the Z0 mass. . . . . . . . . . . . 154
7.4 Doubly charged Higgs particle exchange diagram. . . . . . . . . . . . . . . . 156
7.5 Muonium to anti-muonium conversion. . . . . . . . . . . . . . . . . . . . . . . 157
7.6 Present and future experiments used to map the running of the weak mixing
angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
xxi
List of Tables
1.1 The E158 dataset boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Integrated asymmetries for Run 1 and Run 2. . . . . . . . . . . . . . . . . . . 26
3.2 BPM contribution to Møller detector resolution. . . . . . . . . . . . . . . . . 31
3.3 Run 1 Estimate of uncertainty on APV due to BPM corrections. . . . . . . . 32
3.4 Run 1 systematic uncertainty on APV due to spot size. . . . . . . . . . . . . 38
3.5 Run 2 systematic uncertainty on APV due to spot size. . . . . . . . . . . . . 38
4.1 Measured linearity of the In, Mid, and Out rings of the E158 calorimeter, for
Run I and Run II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Polarimetry uncertainties, relative to Pbeam. . . . . . . . . . . . . . . . . . . . 64
4.3 Pion detector asymmetry results. . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Vertical beam polarization at the target. . . . . . . . . . . . . . . . . . . . . . 78
4.5 Synchrotron asymmetry correction for the Møller detector. . . . . . . . . . . 79
5.1 Contributions to the luminosity monitor asymmetry resolution. *Target boil-
ing is covered in Section 5.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 List of target boiling data runs. . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.1 Regression coefficients for Run I. . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Beam corrections to ARaw of the Møller detector. . . . . . . . . . . . . . . . . 120
xxii
6.3 Comparison of regression and dithering results for the Møller detector asym-
metry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Dilution factors f due to background eP scatters. R is the ratio of elastic to
inelastic eP signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.5 Diluted asymmetries fA due to the electron-proton scattering background.
All entries are in ppb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.6 Regression coefficients of three Mid detector weighting schemes. . . . . . . . 127
6.7 Sensitive pattern-weighted detectors for the regression beam parameters. . . 128
6.8 Run I first-order systematic uncertainties in the regression corrections to the
raw Møller detector asymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.9 Run II first-order systematic uncertainties in the regression corrections to the
raw Møller detector asymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.10 The higher-order systematic uncertainty computed for the Møller detector,
comparing the Møller detector (1) and the Out ring (2), assuming Equation 6.26.133
6.11 The higher-order systematic uncertainty computed for the Møller detector,
comparing the Møller detector (1) and the Out ring (2), using Equation 6.29. 134
6.12 The computed shifts in AMeasured due to the dipole amplitude from transverse
beam polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.13 Synchrotron correction calculated with the SLM. . . . . . . . . . . . . . . . . 138
6.14 The dilution factors f are dimensionless, while the asymmetry corrections fA
are given in units of ppb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.15 The dilution factors f are dimensionless, while the asymmetry corrections fA
are given in units of ppb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.16 The parity-violating asymmetry in Møller scattering. . . . . . . . . . . . . . . 141
xxiii
6.17 Comparison of the regression coefficients for two individual lumi channels and
the full detector. Channel 00 is at the top of the lumi ring, while channel 02
is on the right side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.18 The composite monitor Out-Mid has enhanced energy sensitivity. . . . . . . . 145
6.19 Regression coefficients of the composite monitors Cn. . . . . . . . . . . . . . 145
6.20 Run I systematic uncertainty estimates for the front lumi. The columns labeled
cComp and clumi contain the dominant regression coefficient for the composite
monitor and the lumi, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 146
6.21 Run II systematic uncertainty estimates for the front lumi. The columns
labeled cComp and clumi contain the dominant regression coefficient for the
composite monitor and the lumi, respectively. . . . . . . . . . . . . . . . . . . 146
6.22 Luminosity monitor asymmetry results. . . . . . . . . . . . . . . . . . . . . . 149
7.1 Analyzing powers computed from simulation. The overall average AP is de-
termined by weighting the entries with the corresponding uncertainty on APV . 152
1
Chapter 1
Introduction
The E158 experiment is designed to be a precision low-energy test of the electroweak theory,
complementing the substantial high-energy results obtained by the SLC and LEP colliders.
This introductory chapter provides a background to the experiment by briefly tracing the
evolution of the understanding of the weak force and its unification with the electromagnetic
interaction in the framework of the Standard Model. In addition, the methodology and
timeline of the E158 experiment are outlined.
1.1 Early Study of the Weak Force
The study of the weak force has its beginning in the year 1900 with the discovery of β
radiation by Becquerel [1]. The process was understood as the decay of an atom in state X
to another state X ′ through the emission of an electron e−:
X → X ′ + e−. (1.1)
The decay results in only two products, so conservation of energy dictates that all β decays
should emit an electron with the same energy.
In 1914, Chadwick made the surprising discovery that the energy of the particles emitted
2
in β decay was not a single value, but a continuum [2]. To many scientists, the result
indicated that β decays simply did not conserve energy. Pauli remedied the situation in
1930 by hypothesizing that there was another particle ν1 emitted in β decay, along with
the electron, carrying off the missing energy [3]:
X → X ′ + e− + ν. (1.2)
The particle had to be very light and weakly interacting to have avoided detection. It was
clear that the ν particle could not interact electromagnetically but only through a new
force, called the weak force.
Fermi dubbed the new particle the “neutrino” and in 1934 incorporated it into a success-
ful theory describing β decay. Figure 1.1 depicts the diagram representing Fermi’s model.
The four particles of the decay interact at a single vertex, with a coupling strength denoted
as GF [4].
Figure 1.1: Fermi’s four-point interaction for β decay.
In 1949, it was realized that GF was identical for many different weak decay processes.
This led Lee, Yang, and Rosenbluth to postulate that all weak interactions are mediated
by a massive boson, named the W− (or its anti-particle, the W+) [5]. Figure 1.2 represents
β decay with the introduction of the W−. For energy scales below the mass of the W ,1Pauli actually named the new particle “neutron,” but that name was taken in 1932 with Chadwick’s
discovery of the neutral partner to the proton in the nucleus. Also, the particle ν is actually called theanti-neutrino ν today.
3
the process depicted in Figure 1.2 is indistinguishable from Fermi’s single-vertex process in
Figure 1.1.
Figure 1.2: The W− incorporated into β decay.
1.1.1 Parity Violation
In the early 1950’s, it was observed that two particles called τ+ and θ+ appeared to be
identical in all respects, except that they decayed to states of opposite parity. Lee and
Yang speculated in 1956 that the τ and θ particles were actually the same particle, with
the weak decay mode violating parity [6]. They pointed out that while there was ample
evidence that strong and electromagnetic interactions conserve parity, there was no such
evidence for weak interactions.
The following year, an experiment by C. S. Wu with polarized 60Co demonstrated un-
equivocally that indeed the weak force did not respect parity [7]. Since that time, parity
violation experiments have been an important probe of the structure of the weak interac-
tions.
1.1.2 Electroweak Unification
In 1961, Glashow presented the first work attempting to unify the weak and electromagnetic
interactions into a single framework [8]. In addition to the weak mediators W±, the theory
predicted a weak neutral current, mediated by the Z0. In 1967, Weinberg and Salam cast
the unified electroweak theory in the form of a gauge theory with spontaneous symmetry
4
breaking to explain the difference in the masses of the weak mediators and the photon [9, 10].
The Glashow-Weinberg-Salam (GWS) theory contains a free parameter, the weak mixing
angle θW , that determines the relative strengths of the electromagnetic coupling ge and the
weak couplings gW and gZ through
gW = ge
sin θWand gZ = ge
sin θW cos θW.
The theory also relates the masses of the weak mediators through
MW = MZ cos θW . (1.3)
The theory was bolstered in 1973 with the discovery of a weak neutral current event in
the Gargamelle bubble chamber at CERN [11]. They observed the interaction
νµ + e− → νµ + e−, (1.4)
which could only be mediated by the Z0 boson.
In 1978, the E122 experiment at SLAC observed the parity-violating asymmetry in
polarized electron scattering from a deuterium target. The results were in agreement with
the GWS theory, and ruled out competing models [12, 13]. The weak mixing angle was
measured to be sin2θW = 0.224 ± 0.020.
The knowledge of sin2θW and GF are sufficient to calculate the masses of the W and
Z0 particles in the GWS theory. Using the result of the E122 experiment, it was found
that the W particle should have a mass of ≈ 80 GeV/c2, while the Z0 should have a mass
of ≈ 90 GeV/c2. In 1983, the W± and Z0 were found at CERN at the predicted energies,
dramatically supporting the GWS electroweak theory [14].
5
1.1.3 LEP and SLC
The SLC at SLAC and LEP at CERN were e+e− colliders that operated in the 1990’s to
test the Standard Model to high precision near the mass of the Z0. With few exceptions,
the results were in excellent agreement with the theoretical predictions. The precision of
the collider results can be appreciated by noting that the weak mixing angle was measured
to be
sin2 θW = 0.23113 ± 0.00015, (1.5)
combining all data [15].
1.2 The Role of the E158 Experiment
As with any coupling constant, renormalization causes sin2θW to run as a function of the
four-momentum transfer Q of an interaction. Figure 1.3 depicts the predicted running of the
weak mixing angle with energy. In order to fully explore the Standard Model, measurements
must be performed at several different Q values. The E158 experiment complements the
work done at the SLC and LEP by operating in an energy range far below the mass of the
Z0. Interference with the dominant electromagnetic diagrams allows low Q2 experiments
unique sensitivity to physics effects beyond the Standard Model.
1.3 E158 Overview
The E158 experiment extracts sin2θW at a Q2 of 0.026 (GeV/c)2 by measuring the parity-
violating asymmetry APV in Møller scattering. The experiment is technically challeng-
ing because the asymmetry is expected to be very small, on the order of -150 parts-per-
billion. The measurement represents the first time that parity violation has been observed
6
Figure 1.3: The running of sin2θW with four-momentum transfer Q.
in electron-electron scattering.
Figure 1.4 presents a general diagram of the E158 apparatus. The experiment utilizes
the 48 GeV polarized electron beam provided by the two-mile linear accelerator at SLAC,
scattering off of unpolarized atomic electrons in a fixed target of liquid hydrogen. The
scattered flux is then integrated with the E158 calorimeter, located in the experimental hall
End Station A. The asymmetry APV is defined as the difference in the scattering rate for
each beam helicity, normalized to their sum, given by
APV =dσdΩ
R − dσdΩ
L
dσdΩ
R+ dσ
dΩ
L, (1.6)
where L and R refer to the helicity of the incident beam.
Figure 1.4: General diagram of the E158 apparatus. The regions of the E158 calorimeterare named as follows: A.) In ring B.) Mid ring C.) Out ring and D.) eP detector.
7
The calorimeter is divided into four annular regions, designated as In, Mid, Out, and
eP. The E158 spectrometer focuses the scattered electrons so that the inner three regions
of the calorimeter are dominated by Møller scattering events, while the outer eP ring is
dominated by electron-proton scatters. The In and Mid region together are known as the
Møller detector2, and are used for the measurement of APV .
1.4 Experiment Timeline
The data for the E158 experiment was collected over four distinct periods, designated as
Run 0 through Run 3. Each data set is analyzed independently to determine sin2θW . The
results of each Run are then combined to obtain the overall E158 result.
Table 1.1 presents the amount of data in each Run. This paper covers the combined
results of Run 1 and Run 2, representing slightly over half of the full E158 data set. The
analysis of Run 3 is still underway.
Dataset Time Total Data (pulses)Run 0 Winter 2001 Engineering RunRun 1 Spring 2002 212 millionRun 2 Fall 2002 236 millionRun 3 Summer 2003 360 million
Table 1.1: The E158 dataset boundaries.
2The Out region could have been included, but was found to be susceptible to large systematic uncer-tainties (Section 6.7.2.2).
8
Chapter 2
Theory
The SLAC E158 experiment measures the parity-violating asymmetry in Møller scattering
at a Q2 five orders of magnitude below the weak scale. The measurement probes the
Standard Model at the one loop level, providing insight into the running of the electroweak
observable sin2θW .
This chapter presents the theoretical prediction for the parity-violating asymmetry APV
in Møller scattering and its relation to the weak mixing angle. Additionally, previous low
Q2 electroweak experiments and their results are described.
2.1 The E158 Experiment
The experiment utilizes a 48 GeV polarized electron beam on unpolarized atomic electrons
in a liquid hydrogen target to measure the parity-violating asymmetry in Møller scattering,
at a Q2 of 0.026 (GeV/c)2. Radiative corrections reduce the tree-level asymmetry by 40%.
The large relative size of higher-order effects allows the experiment to be a sensitive probe of
the Standard Model. Sections 2.3 and 2.4 present the theoretical calculation of the expected
asymmetry.
The effect from radiative corrections can be neatly accommodated by defining a Q2
dependent weak mixing angle. Figure 2.1 displays the running of sin2θW (Q2) as well as
9
the results of several precision electroweak experiments. The NuTeV point refers to a
Figure 2.1: The running of sin2θW with Q.
neutrino experiment conducted at Fermilab [16], and the APV point represents atomic
parity violation studies performed by NIST and the University of Colorado [17]. These
experiments are described in Sections 2.2.1 and 2.2.2.
2.2 Previous Low Q2 Electroweak Measurements
It is clear from Figure 2.1 that the Z-pole experiments have measured sin2θW to high
precision, and the results are in agreement with the Standard Model. In contrast, the low
Q2 regime has been probed by only two experiments, with much less precision. The lack of
electroweak measurements for low Q2 is the primary motivation for the E158 experiment.
Both the NuTeV experiment and the APV measurements require considerable theoreti-
cal input to extract the weak mixing angle. The E158 experiment is designed to complement
these experiments by examining the comparatively clean process of Møller scattering. Also,
the energy scale is between the previous measurements, allowing the E158 experiment to
provide a unique point on the sin2θW (Q2) curve.
10
2.2.1 NuTeV Experiment Overview
The NuTeV experiment compared neutrino and anti-neutrino scattering rates to determine
sin2θW [16]. The high purity neutrino beams were delivered by the Fermilab accelerator,
and cross sections were measured in a 120-foot-long steel detector. The relevant Feynman
diagrams are shown in Figures 2.2 and 2.3. The Q2 of the experiment was 20 (GeV/c)2.
Figure 2.2: Neutrino scattering diagrams.
Figure 2.3: Anti-neutrino scattering diagrams.
The weak mixing angle is extracted from the data by constructing the Paschos-Wolfenstein
ratio R− [18], defined as
R− ≡ σ(νµN → νµX) − σ(νµN → νµX)σ(νµN → µ−X) − σ(νµN → µ+X)
. (2.1)
The quantity R− is directly related to the weak mixing angle through
R− =12− sin2 θW . (2.2)
11
The NuTeV Collaboration reports a value for sin2θW that is 3σ above the Standard Model
prediction.
Many attempts have been made to reconcile the NuTeV result within the framework of
the Standard Model. The primary focus has been on understanding the parton distribution
functions used in the determination of R−. Nuclear effects are complex, and it is possible
to shift the value of sin2θW based on the assumptions of these distributions [19]. The
induced shifts have not yet been found to align significantly the NuTeV result with the
Standard Model prediction. However, a recent O(α) re-analysis of deep-inelastic neutrino
scattering indicates that perhaps the theoretical uncertainties used in the extraction of
sin2 θW are enough to reconcile the 3σ shift in the NuTeV result. The work in this area is
still ongoing [20].
2.2.2 Atomic Parity Violation Overview
Electroweak experiments in atomic physics measure the perturbation of electronic orbitals
induced by Z0 exchange (Figure 2.4). The potential of the nucleus can be described as due
to the standard electric charge Z and the weak charge QW . The Q2 is very low, set by the
energy scale of the atomic orbitals.
Figure 2.4: Tree-level contributions to the nuclear potential.
The Z0 diagram produces small mixings of the unperturbed orthogonal electron orbitals.
The overlap allows otherwise forbidden transitions to occur. In principle, atomic parity
12
violation experiments measure transition rates among these states to obtain QW .
At tree level, the weak charge of the nucleus is given by
QW = −N + Z(1 − 4 sin2 θW ), (2.3)
where N is the number of neutrons and Z is the number of protons [21]. Since sin2θW
is numerically very close to 0.25, Equation 2.3 essentially reduces the weak charge to the
number of neutrons N . Radiative corrections alter the tree-level prediction in Equation 2.3,
making atomic parity violation experiments sensitive to the running of the weak mixing
angle [22, 23].
Because the weak charge is proportional to the number of neutrons in the nucleus, heavy
atoms are the preferred subjects in APV experiments. However, to extract sin2θW from a
measurement, it is necessary to have a precise model of the electronic wavefunctions involved
in the transitions observed. The determination of the wavefunctions in heavy atoms is
complex and is the greatest source of uncertainty for atomic parity violation measurements.
The most precise APV experiment utilized 133Cs [17]. Cesium is a good subject because
it is a heavy atom with a single valence electron. The experiment initially reported a weak
charge that was 2.5σ below the Standard Model prediction. However, subsequent complex
electron wavefunction analyses [24, 25, 26, 27, 28, 29] have moved this value to within 0.5σ
of the theoretical prediction. The most recent result is plotted in Figure 2.1.
13
2.3 E158 Experiment at Tree Level
The E158 experiment measures the parity-violating asymmetry APV in Møller scattering.
The asymmetry is defined as
APV ≡dσdΩ
R − dσdΩ
L
dσdΩ
R + dσdΩ
L, (2.4)
where dσdΩ is the differential cross section, and L and R refer to the helicity of the electron
beam. The target electrons are unpolarized. At tree level, there are four diagrams which
contribute to APV , depicted in Figure 2.5. In the limit that m2e− << Q2 << m2
Z0, the
+ Crossed Diagrams
Figure 2.5: Tree-level contributions to APV .
asymmetry is given by [30]
APVTree =−GµQ2
√2πα
1 − y
1 + y4 + (1 − y)4(1 − 4 sin2 θW ), (2.5)
where
y ≡ 1 − cos θCM2
. (2.6)
Gµ is the Fermi constant obtained from the muon lifetime formula [31], and the fine structure
constant α is roughly 1137 , as is suitable for low Q2 measurements. The θCM term refers to
the center-of-momentum scattering angle.
Because sin2θW ≈ 14 , APV is very sensitive to the weak mixing angle. The relation is
14
made clear by noting
∆APV
APV=
∆ sin2 θW14 − sin2 θW
. (2.7)
The final E158 result is expected to measure APV to the precision of 12%, corresponding
to a determination of sin2θW to ±0.0014.
It is important to note that the tree-level asymmetry given in Equation 2.5 is a very
small number. Even at its maximum (y = 12 ), APVTree is only -300 ppb (parts-per-billion).
It is the smallness of this number that presents the greatest challenge to the success of the
experiment.
2.4 Radiative Corrections to APV
Because the tree-level asymmetry is suppressed by a factor of 1−4 sin2 θW , the contribution
of higher-order diagrams is effectively enhanced. This feature is responsible for the running
of the weak mixing angle in Figure 2.1, and allows the E158 experiment to be a sensitive
probe of the Standard Model.
Marciano has evaluated APV including one-loop radiative corrections and found the
corrected asymmetry to be given by [32]
APV =−ρGµQ2
√2πα
1 − y
1 + y4 + (1 − y)4×
1 − 4κ(0) sin2 θW +α
4π sin2 θW
− 3α(1 − 4 sin2 θW )32π sin2 θW cos2 θW
[1 + (1 − 4 sin2 θW )2] + F1(y,Q2) + F2(y,Q2),
(2.8)
where
sin2 θW ≡ sin2 θMSW (M2
Z0). (2.9)
15
The term in the braces now has a Q2 dependence, producing the running of sin2θW depicted
in Figure 2.1. The precise definition of sin2θW (Q2) is given in Section 2.4.5.
The one-loop corrections reduce the tree-level prediction for APV by ≈ 40%. The
expected asymmetry for the E158 experiment is then roughly -180 ppb, for y = 12 . The
following sections describe the sources and sizes of the terms in Equation 2.8.
Because the effect of higher-order diagrams is large for low Q2, the E158 experiment
can be used as a sensitive probe for physics effects beyond the Standard Model at the TeV
scale. Section 7.2 describes the new physics limits that can be set with the experiment, as
well as presenting the current limits set by previous experiments.
2.4.1 γ − Z0 Mixing Diagrams
The largest one-loop corrections are contained in the κ(0) term in Equation 2.8. The
relevant processes are represented by γ − Z0 mixing diagrams and the W contribution to
the electron anapole moment, depicted in Figure 2.6. The Q2 dependence of these diagrams
is contained in F2(y,Q2).
+ Inverted + Crossed Diagrams
Figure 2.6: Photon-Z0 mixing diagrams and the W contribution to the anapole moment.
The fermionic loop in the first diagram in Figure 2.6 presents the greatest calculational
16
challenge. The quark contribution cannot be evaluated perturbatively and must be deter-
mined from e+e− →hadrons experimental data. The uncertainty on κ represents the largest
contribution to the theoretical uncertainty on APV . Evaluating the diagrams, κ is found to
be
κ(0) = 1.0301 ± 0.0025. (2.10)
The 3% correction to sin2θW by κ(0) corresponds to a 37% reduction in APV . It should be
noted here that F2(12 , 0.026 (GeV/c)2) = 0.00002, a negligible contribution.
2.4.2 Heavy Box Diagrams
Box diagrams containing heavy bosons comprise the next correction to APVTree. These dia-
grams are depicted in Figure 2.7. The W diagram contributes the α4π sin2 θW
term in Equa-
+ Crossed Diagrams
Figure 2.7: Heavy boson box diagrams.
tion 2.8. It yields a 4% increase in APV over the tree-level expression. The Z diagrams
produce the −3α(1−4 sin2 θW )32π sin2 θW cos2 θW
[1 + (1 − 4 sin2 θW )2] term in Equation 2.8, resulting in a neg-
ligible 0.1% shift to the tree-level expression.
2.4.3 γ − Z0 Box Diagrams
The final contribution to the one-loop corrected value of APVTree comes from box diagrams
containing both a photon and a Z0 exchange. The relevant diagrams are depicted in Fig-
17
ure 2.8. The Z0 contribution to the electron anapole moment also contributes at this level,
shown in Figure 2.9.
+ Crossed Diagrams
Figure 2.8: Photon-Z0 box diagrams.
+ Inverted + Crossed Diagrams
Figure 2.9: Anapole moment contribution from the Z0.
The contribution of these diagrams is contained in the F1(y,Q2) term in Equation 2.8.
Evaluating F1 for the values appropriate for the experiment, it is found that
F1
(12, 0.02 (GeV/c)2
)= −0.0041 ± 0.0010. (2.11)
The F1 function constitutes a 6% reduction in APVTree, largely canceling the previous W box
diagram term.
18
2.4.4 ρ Term
The ρ term in Equation 2.8 follows from the convention chosen for Gµ and renormalization
of the Z amplitude [33]. The explicit form is
ρ = 1 +α
4π×
34 sin4 θW
log cos2 θW − 74 sin2 θW
+3
4 sin2 θW
m2t
m2W
+ (2.12)
3m2H
4 sin2 θWm2Z
log cos2 θW
m2Z
m2H
cos2 θW − m2H
m2Z
+1
cos2 θW
log m2H
m2Z
cos2 θW − m2H
m2Z
,
where mt refers to the mass of the top quark, and mH denotes the mass of the Standard
Model Higgs. The dependence on these masses is very slight. The other masses, mZ and
mW , refer to the standard weak gauge bosons. Assuming mt = 170 GeV/c2 and mH =
200 GeV/c2, one finds ρ = 1.00122, a totally negligible correction to the overall asymmetry.
2.4.5 Definition of sin2θW (Q2)
The precise definition of sin2θW (Q2) is a matter of convention. Often, only the terms due
to the γ-Z mixing and the W contribution to the electron anapole moment (Section 2.4.1)
are grouped into the definition of the weak mixing angle through
1 − 4 sin2 θMSW (Q2) ≡ 1 − 4κ(0) sin2 θMS
W (M2Z) + F2(Q2). (2.13)
The definition established in Equation 2.13 is typically preferred by theorists, and was used
to produce Figure 2.1.
On the other hand, experimentalists usually report an “effective” weak mixing angle.
19
This amounts to defining sin2θW (Q2) so that the tree-level asymmetry formula holds, with
1 − 4 sin2 θeffW (Q2) ≡ (2.14)
ρ
1 − 4κ(0) sin2 θW +
α
4π sin2 θW
− 3α(1 − 4 sin2 θW )32π sin2 θW cos2 θW
[1 + (1 − 4 sin2 θW )2]
+F1(y,Q2) + F2(y,Q2)
Due to cancellations, the two definitions are very nearly equal for low Q2. For the parameters
of the E158 experiment, one finds sin2 θMSW = 0.2381 and sin2 θeffW = 0.2385.
20
Chapter 3
E158 Beamline and BeamMonitoring
This chapter describes the major components of the E158 apparatus, from the beginning
of the accelerator up to the final collimation before the detectors. The polarized beam,
precision beam monitors, the liquid hydrogen target, and the spectrometer are covered.
3.1 Polarized Source
The helicity of the primary electron beam is controlled at the polarized source, located
upstream of the linac (Figure 3.1). The source houses a complex optical system, depicted
in Figure 3.2, which is employed to produce high beam polarization while minimizing sus-
ceptibility to helicity-correlated systematic effects [34, 35]. The following sections detail the
components of the optical system and their relevance to the E158 experiment.
Figure 3.1: Location of the polarized source room.
21
Figure 3.2: E158 optics configuration at the source.
3.1.1 Laser Bench
Laser light production and pulse-shaping occur on the laser bench. The system begins with
a Flash:Ti laser which generates 12 µs pulses of linearly polarized light. A Brewster tuner
is utilized to control the wavelength of the laser, holding it to within 4 nm of the central
854 nm wavelength. At 120 Hz, the laser power is roughly 2 W.
The Slice Pockels cell is used to sample the 300 ns portion of the laser pulse with the
most favorable characteristics, balancing intensity with jitter. The Slice cell is a piezoelectric
crystal, with optical properties that are affected by the applied voltage. At zero Volts the
crystal is optically neutral, while at 3000 Volts it functions as a half-wave plate. The
Slice cell sits between two crossed linear polarizers. Biasing the Slice as a quarter-wave
plate allows the laser light to pass through both polarizers. When the cell is unbiased, an
extinction ratio of 500 is achieved.
The Tops Pockels cell is used to shape the time profile of the laser pulse, to match the
properties of the cathode. The Tops cell is placed between aligned linear polarizers and is
22
pulsed at low voltage to produce mild light extinction.
3.1.2 Combiner Bench
The elements on the combiner bench are only used for diagnostics. The harmonic beam
splitter (HBS) diverts roughly 2% of the laser light to two separate beam monitors. The
spectrometer records the laser wavelength, while the photodiode is used to monitor laser
power.
3.1.3 Wall Bench
The components on the wall bench are used to define the helicity of the electron beam and
to suppress helicity-correlated beam asymmetries. The intensity asymmetry (IA) Pockels
cell is part of a feedback system designed to reduce the charge asymmetry of the electron
beam. Analogous to the Tops cell, the IA cell is operated at low voltage between aligned
linear polarizers. It is pulsed based on charge measurements performed early in the linac.
The circular polarizer (CP) and phase shifter (PS) Pockels cells are responsible for
defining the helicity of the beam that ultimately reaches the cathode. The CP cell is
pulsed at its quarter-wave voltage, converting the incoming light from linear to circular
polarization. The helicity of the light is reversed by changing the sign of the voltage bias.
The PS cell is run at lower voltages and is used primarily to correct for residual linear
polarization left by the CP cell.
The piezomirror is used to reduce helicity-correlated beam position asymmetries. Like
the IA Pockels cell, the piezomirror is controlled based on beam measurements using mon-
itors early in the linac. The voltage applied to the mirror changes the angle of reflection,
ultimately moving the laser spot on the cathode.
23
The asymmetry inverter is used to combat helicity-correlated effects due to asymmetries
in the laser profile on the cathode. The inverter lenses can be moved as a unit between two
configurations. The settings complement each other, with one spatially inverting the beam
profile compared to the other. The lenses were toggled once in Run 1 and once in Run 2.
Following the wall bench, the laser light travels down the Optical Transport System
(OTS) to the cathode. The OTS is essentially a 20 m pipe, filled with nitrogen, linking the
source room with the cathode room. It contains several lenses configured to preserve the
quality of the beam.
3.1.4 Polarized Gun
The gun bench holds the final optics preceding the cathode. The lenses on the bench are
configured as a telescope, used to match the laser spot size to the dimensions of the cathode.
A mirror can be inserted after the lenses to divert the laser beam to a diagnostic target.
Optically, the target is in the same position as the real cathode. Centering light on the
target ensures that the light will be centered on the cathode when the mirror is removed.
The target is monitored remotely by a camera.
The final component on the bench is an insertable half-wave plate. It is used to combat
helicity-correlated systematic effects by reversing the helicity of the laser light defined with
the CP cell, while leaving the rest of the system unchanged. The wave plate is toggled once
every two days during E158 production running.
The cathode used for the E158 experiment is composed of a strained GaAs lattice [36].
This type of cathode has been found to provide the highest presently achievable beam
polarization along with high current. The polarization of the E158 electron beam was
measured to be ∼85% (Section 4.2), with no evidence that the cathode was charge limited.
24
3.1.5 Helicity Sequence
The helicity of the electron beam is defined in sets of four pulses. The helicity of the first
pair is chosen randomly, while the second pair is the conjugate of the first. Each quad of
pulses then contains two separate sets of pulses, with the first pulse paired with the third
and the second pulse paired with the fourth. Figure 3.3 illustrates the pulse sequence.
Figure 3.3: Helicity of the E158 electron beam.
Pairs of pulses, rather than single pulses, are the fundamental unit of the experiment.
Quantities of the form
PulseR − PulseL
PulseR + PulseL(3.1)
are called asymmetries, where L and R refer to the helicity state. The results from the
detectors are reported as asymmetries. The units appropriate for the E158 experiment are
ppm or ppb, signifying parts-per-million or parts-per-billion. Quantities of the form
PulseR − PulseL (3.2)
are called differences, but may also be referred to as asymmetries. Beam position monitor
results are reported in this form.
25
3.1.6 Beam Asymmetries
The polarized source is configured to decrease the size of helicity-correlated beam asymme-
tries. For a full E158 dataset, careful calibration of the positive and negative CP cell bias
voltages suppresses the expected intensity asymmetry from ∼1000 ppm down to ∼100 ppm.
The intensity asymmetry feedback then reduces the asymmetry to ∼100 ppb [35]. The large
suppression is critical for controlling systematic uncertainties because APV is only ≈ -150
ppb.
Figure 3.4 depicts the integrated charge asymmetry versus time for the toroid used for
the intensity feedback. The horizontal scale covers all of Run 1 and Run 2. The dotted line
indicates purely statistical scaling. The final asymmetry is at the level of a few hundred
ppb, as expected with the feedback asymmetry suppression.
Figure 3.4: Integrated charge asymmetry measured near the source, spanning all of Run 1and Run 2.
While the intensity feedback ensures that the charge asymmetry near the beginning of
the linac is suppressed, it is the asymmetry at the target, more than two miles away, that
is the relevant quantity for the E158 analysis. Figure 3.5 depicts the charge asymmetries
measured by the toroids just upstream of the target for both Run 1 and Run 2. The asym-
metry suppression due to the intensity feedback clearly translates into the target region,
26
Run 1Parameter Integrated Asymmetry
Charge 210 ± 319 ppbEnergy -0.1 ± 1.4 keV
X -16.3 ± 5.6 nmY -3.0 ± 4.0 nm
X Angle 0.38 ± 0.23 nRY Angle 0.11 ± 0.07 nR
Run 2Parameter Integrated Asymmetry
Charge 496 ± 335 ppbE 0.9 ± 2.1 keVX 13.0 ± 6.7 nmY -15.9 ± 5.2 nm
X Angle 0.33 ± 0.22 nRY Angle 0.13 ± 0.11 nR
Table 3.1: Integrated asymmetries for Run 1 and Run 2.
although the convergence is not as strong as at the source.
Figure 3.5: Run 1 and Run 2 target region integrated charge asymmetry.
Since most beam parameters are correlated to charge, the intensity asymmetry feedback
also suppresses position asymmetries. (The position feedback with the piezomirror is also
employed, though it proved to be less effective than the IA feedback.) In the absence of
feedbacks, position asymmetries at the ∼100 µm would be expected for an E158 data set.
The feedbacks suppress the asymmetries to the level of ∼1 nm. Table 3.1 presents the
integrated asymmetries for Run 1 and Run 2.1 The asymmetry suppression observed is at
the level expected for the experiment [37].1The average asymmetry is computed by weighting the data with the resolution of the primary E158
detector. The asymmetries are then directly applicable to the analysis of the detector results.
27
3.2 Beam Position Monitors
The position of the electron beam is measured with RF cavity beam position monitors
(BPMs). Figure 3.6 depicts the location of the E158 BPMs. The X and Y positions of the
beam at the target are measured with the target BPMs. The X and Y angles are computed
using the difference between the target and angle BPMs, which are separated by 40 meters.
The energy BPMs are located in an area of high dispersion so that a horizontal position
measurement actually corresponds to an energy determination. The three BPMs located
close to the source are employed for the source position feedback.
Figure 3.6: Location of E158 beam monitors.
3.2.1 BPM Operation
The beam position monitors are composed of three separate cavities [38]. Figure 3.7 is
a picture of an E158 BPM. When the electron beam traverses the device, it excites the
resonant electromagnetic modes of the cavities. The amplitude of the response is picked up
by an antenna in each cavity and read out as the signal.
The rectangular cavities are employed to measure X or Y position. The beam excites
either the TM210 or TM120, with an amplitude that is proportional to both beam position
and charge. The position cavities are then normalized to beam charge with the E158 toroids
(Section 3.3), leaving only the sensitivity to position.
28
The final BPM cavity is cylindrical, and is only sensitive to beam charge. In principle,
it could be used for the normalization of the BPM position cavities. The toroids were used
instead because they have superior charge asymmetry resolution.
Figure 3.7: E158 beam position monitor.
Each E158 beam pulse is roughly 300 ns long. However, the pulse itself is composed
of smaller electron bunches, grouped at the 2856 MHz rate of the accelerator. The BPM
cavities respond to the bunches, producing a signal that oscillates at the accelerator rate.
The resonant frequency of the cavities is 2856 ± 0.3 MHz, designed to optimize the response
of the device.
The oscillation in the output signals is removed by mixing the BPM signals with an
auxiliary signal locked to the accelerator rate. The mixing electronics produce two outputs,
with amplitudes proportional to the amount of signal in phase or out of phase with the aux-
iliary signal. The mixer is tuned so that one of the output signals is maximized, minimizing
the other. A feedback system was employed to ensure that phase drifts are counteracted
and the primary signal remains a maximum.
Figure 3.8 depicts the two signals returned by the mixer for one of the BPM cavities.
The signal quickly increases with the 300 ns beam pulse and then exponentially decays.
29
Figure 3.8: BPM signals after mixing.
The signals are read into custom built 16-bit VME ADCs (analog to digital converters)
as part of the data stream. High-resolution ADCs are required to match the resolution of
the BPMs.
3.2.2 BPM Performance
The BPMs are deployed in pairs or triplets (Figure 3.6) to allow for cross-checks between
devices in close proximity. The performance of a BPM pair is quantified by the distribution
of the agreement δ, defined as
δi ≡ 12(∆BPM1
i − ∆BPM2i ), (3.3)
where ∆BPMni indicates the position difference measured with BPMn for the ith pulse
pair. The width of the distribution is the pulse-pair resolution of the BPM pair. Figures 3.9
and 3.10 depict typical resolutions of the E158 BPMs.
The resolution of each of the BPMs exceeds the requirements of the E158 experiment.
This can be demonstrated by noting that the asymmetry measured with the Møller detector
30
Figure 3.9: Angle and position BPM resolutions.
is corrected for beam asymmetries by
APV = ARaw −5∑
n=1
cn∆BPMn, (3.4)
where APV is the true physics asymmetry, ARaw is the measured detector asymmetry,
∆BPMn is the position difference measured with BPMn, and cn is the experimentally
determined sensitivity of the detector to beam motion. The sum covers the beam parameters
of energy, X and Y position, and X and Y angle. This method is called regression, and is
discussed in detail in Section 6.3. The resolution of the BPMs contributes to the resolution
31
Figure 3.10: Energy BPM resolutions.
Parameter Detector Coefficient c BPM Resolution Resolution ContributionX -0.1 ppm/µm 2.4µm 0.2 ppmY -1.3 ppm/µm 3.6µm 4.7 ppm
Angle X -52.6 ppm/µR 0.13µR 6.8 ppmAngle Y 12.1 ppm/µR 0.15µR 1.8 ppmEnergy -15.2 ppm/MeV 1.1 MeV 16.7 ppmTotal 18.7 ppm
Table 3.2: BPM contribution to Møller detector resolution.
of the detector σDetector through
σ2Detector =
5∑n=1
(cnσBPMn )2, (3.5)
where σBPMn is the resolution of BPMn. The contributions to the resolution of the Møller
detector are presented in Table 3.2. The resolution of the detector is typically around 200
ppm, while the contribution due to BPM resolution is negligible, at less than 20 ppm.
The resolution of the BPMs is also constrained by the general goal that individual
contributions to the systematic uncertainty on APV be 5 ppb or less, over a full dataset.
The uncertainty on APV measured with the Møller detector due to the BPM corrections is
σPVA =5∑
n=1
cn < δn >, (3.6)
32
Parameter Detector Coefficient c BPM Agreement δ σsystematicAPV
X 0.4 ppm/µm 1.0 ± 0.6 nm 0.4 ± 0.2 ppbY -1.2 ppm/µm 0.0 ± 1.0 nm 0.0 ± 1.2 ppb
Angle X -66.1 ppm/µR -0.07 ± 0.05 nR 4.6 ± 3.3 ppbAngle Y 7.3 ppm/µR 0.02 ± 0.03 nR 0.1 ± 0.2 ppbEnergy -25.9 ppm/MeV 0.0 ± 0.2 keV 0.0 ± 5.2 ppb
Table 3.3: Run 1 Estimate of uncertainty on APV due to BPM corrections.
where < δn > is the average BPM agreement for the nth BPM pair2. Table 3.3 details the
observed BPM agreement for Run 1 and the contribution to the uncertainty on APV [39].
The uncertainty on the agreement is proportional to the resolution of the BPMs.
Each uncertainty on σsystematicAPV is near the goal of 5 ppb or below for all beam parameters,
implying that the BPM resolution is adequate for the experiment.
3.3 Charge Monitors
The beam charge is measured by several toroids distributed along the beamline (Figure 3.6).
The toroids located near the source are used for the intensity feedback, while the toroids near
the target are used for normalizing the detector signals. Specifically, the raw asymmetry
ARaw measured with the Møller detector is
ARaw =
(MT
)R −(MT
)L(MT
)R+(MT
)L , (3.7)
where M and T refer respectively to the Møller detector and toroid signals, and L and R
refer to the helicity of the beam.2It will be demonstrated in Section 6.7.2.2 that this is a naive estimate. For the present purposes,
however, it is adequate.
33
3.3.1 Toroid Overview
Each toroid is composed of an iron ring wrapped with copper wire, positioned around
ceramic portions of the beam pipe. When a charged particle passes through the ring,
a voltage is produced in the wire by inductance. The signal is then amplified and sent
through a rectifier before being read into the ADCs as the charge measurement.
3.3.2 Toroid Performance
Because the toroid signal is used for normalizing the detector signal, it is desirable that the
toroid intensity asymmetry resolution be much better than the Møller detector resolution
of 200 ppm.
The toroid resolution is determined experimentally in a fashion similar to the method
employed for the BPMs in Equation 3.3. The resolution is defined as the width of the
distribution of the agreement δi, given by
δi =1√2(A1
i −A2i ). (3.8)
The Ani terms refer to the asymmetry measured by toroid n for the ith beam pulse. The
prefactor is 1√2, instead of 1
2 used for the BPMs because only one toroid is used for the
charge normalization, instead of a pair.
Figure 3.11 presents the distribution of the agreement δ for a typical one-hour data run.
The width is 59 ppm, which is well below the 200 ppm resolution of the Møller detector.
The average resolution during the experiment was actually closer to 50 ppm, while periods
as low as 30 ppm were observed. The drifts in resolution appear to be coming from the
toroid amplifying electronics.
34
Figure 3.11: Toroid resolution.
The toroids are also required to have a signal response as linear as the detectors, because
they are used for direct normalization. Section 4.1.4 demonstrates that the Møller detector
is linear to the level of roughly 99%, so it is required that the toroid be linear to that level
as well.
Each toroid mounted on the beamline has a single wire passing through its ring that can
be used for in situ calibration and linearity testing. A calibrator which can produce pulses
with better than 0.05% charge stability is used to test the response of the toroid to several
different currents. The curvature in the plot of the toroid versus the calibrator determines
the linearity.
Figure 3.12 presents the results of a toroid linearity test. The error bars on the plot,
due to calibrator fluctuations, are much smaller than the data points themselves. Also note
that the toroid is normalized so that its slope versus the calibrator is unity. The coefficient
of the second order term in the fit is quite small, indicating that the toroid is linear to
better than 99%, as required.
35
Figure 3.12: Toroid signal versus calibrator signal.
3.4 Wire Array
The transverse spatial profile of the beam is monitored with a device called the wire array.
It is positioned directly upstream of the target (Figure 3.6) and can be remotely moved in
and out of the beam path. It is composed of two sets of 48 parallel wires, one horizontal and
one vertical, forming a grid through which the beam passes. The wires are composed of a
copper-beryllium alloy, with a diameter of 180 microns and a spacing of 356 microns. When
the beam passes through the grid, it causes the wires to lose electrons, producing a small
voltage which is read out as the signal. Foils near the wires are held at positive potential to
attract the liberated electrons to reduce the chance for recombination, enhancing the signal
by a factor of two. Figure 3.13 presents a typical profile measurement provided by the wire
array. The beam shape is roughly Gaussian in both axes, with an RMS of approximately 1
mm.
In addition to monitoring the shape of the beam, the wire array is used to measure the
helicity-correlated spot size asymmetry. The size and uncertainty of the spot size asymmetry
36
Figure 3.13: Wire array profiles for both X and Y axes.
have implications for the systematic uncertainty on APV measured with the Møller detector.
The detector sensitivity to spot size α can be expressed as
AMeasured = APV + α∆S, (3.9)
where AMeasured is the asymmetry measured with the detector, APV is the true physics
asymmetry, and ∆S is the spot size asymmetry. The coefficient α can be experimentally
determined by comparing the widths of the distributions of the hybrid asymmetries
J+ ≡ AMeasured + η∆S (3.10)
and
J− ≡ AMeasured − η∆S. (3.11)
The η term is an arbitrary scale factor inserted for dimensional concerns, and also functions
to make the detector and wire array contributions to J± roughly the same size. In the
absence of spot size sensitivity, the width of the distributions of J+ and J− would be the
37
same. The degree to which they differ determines the size of α through
α =σ2J+
− σ2J−
4ησ2∆S
. (3.12)
Figure 3.14 depicts the spot size sensitivity determined with this method for each de-
tector3, for both Run 1 and Run 2. Because the coefficients differ among the detectors,
the sensitivity is dominated by geometry rather than a common effect, such as spot size
induced target density fluctuations. In all cases, the magnitude of the sensitivity is greater
in Run 1 than Run 2, most likely due to different beam tunes.
Figure 3.14: Detector spot size correlation coefficients.
The coefficients can be combined with the helicity-correlated spot size differences mea-
sured with the wire array to produce the systematic uncertainty due to spot size on the
asymmetry measured with the detectors. Tables 3.4 and 3.5 detail the computed uncertain-
ties for Run 1 and Run 2.
The third column in the tables represents the contribution to the asymmetry measured3The luminosity monitor is a low-angle detector used for a null-asymmetry measurement. It is described
in Chapter 5.
38
Average Spot size Asymmetry Systematic Uncertainty EstimateDetector α (ppm/mm2) < ∆S > (10−6mm2) α < ∆S > (ppb)Møller -61 ± 4 -0.3 ± 0.4Out 546 ± 17 5.5 ± 6.9 3.0 ± 3.8
Front Lumi 138 ± 6 0.8 ± 1.0Back Lumi 227 ± 9 1.2 ± 1.6
Table 3.4: Run 1 systematic uncertainty on APV due to spot size.
Average Spot size Asymmetry Systematic Uncertainty EstimateDetector α (ppm/mm2) < ∆S > (10−6mm2) α < ∆S > (ppb)Møller 19 ± 3 -0.2 ± 0.5Out 253 ± 32 -3.7 ± 25.2 -0.9 ± 6.4
Front Lumi 72 ± 8 -0.3 ± 1.8Back Lumi 73 ± 7 -0.3 ± 1.8
Table 3.5: Run 2 systematic uncertainty on APV due to spot size.
with the detectors due to spot size. Because α < ∆S > is consistent with zero in all cases,
the uncertainty on this number will be used as the estimate for the spot size systematic
uncertainty for the detectors. The Run 2 uncertainty is larger because the wire array was
inserted for less data compared to Run 1.
3.5 Skew Quadrupole Magnet
The emission of synchrotron radiation traversing the A-Line bend leading up to the exper-
imental hall has the effect of actually improving the quality of the beam in the horizontal
direction [40]. Higher energy electrons emit more radiation while lower energy electrons emit
less, pushing all electrons toward the average energy. Because virtually all of the bends are
in the horizontal plane, there is no enhancement for the vertical beam parameters.
The E158 experiment introduced a skew quadrupole magnet near the end of the A-
Line to mix the vertical and horizontal beam parameters. The location of the magnet is
depicted in Figure 3.15. The skew quad is a standard quadrupole magnet rotated around
the beamline by 45o. By mixing the horizontal and vertical beam parameters, the overall
39
quality of the beam is improved.
Figure 3.15: Location of skew quadrupole magnet.
Figure 3.16 presents the effect of the skew quad on the detector asymmetry distributions.
The width of the distribution is the resolution of the detector. The presence of the skew
quad dramatically improves the resolution of the luminosity monitor and the Out detector,
while having little effect on the Møller detector.
Figure 3.16: Effect of the skew quad on detector resolution.
The presence of the skew quad was not found to affect the beam position jitter or energy
40
jitter in any discernible way. The resolution enhancement for the Out and lumi detectors
is likely a geometric effect related to the beam divergence.
3.6 Liquid Hydrogen Target
The E158 target is a 1.5-meter-long cylinder of liquid hydrogen [41]. Hydrogen was chosen
because the background electron-proton scattering events are more easily separated and
modeled than for other targets.
The hydrogen in the target is maintained at a temperature of 20 K. The density is
0.07 g/cm3, making the target 0.17 radiation lengths. The hydrogen is continually pumped
around the target loop at 10 m/s, to minimize density changes due to heating by the primary
beam. Mesh discs are positioned in the target cell, out of the path of the beam, to induce
turbulent flow and further reduce potential density fluctuations. Figure 3.17 depicts the
components of the target loop.
Figure 3.17: E158 cryotarget loop.
The hydrogen is cooled by 14 K helium gas flowing through the heat exchanger. The
helium is provided by a refrigerator located in a building adjacent to End Station A. The loop
41
itself is wrapped in 30 layers of aluminized Kapton, to minimize heating due to radiation.
The entire loop sits inside a vacuum chamber, depicted in Figure 3.18. The chamber is
large enough to allow the entire target loop to be retracted out of the beamline.
Figure 3.18: Cryotarget scattering chamber.
The electron beam deposits 700 W of power in the target for the highest current and
repetition rate used by the E158 experiment. The temperature of the target is stabilized
through an adjustable heater located at the end of the heat exchanger [42]. The target
control program automatically monitors the beam current and rate, and adjusts the heater
to maintain stable running.
The target loop contains 55 liters of hydrogen, with 25 liters in the target cell. The full
loop has an explosive yield equivalent to 8 kg of TNT, a serious safety concern. The target
loop is equipped with burst-discs which rupture when the target loop pressure exceeds safe
running conditions. The hydrogen is then directed out of the End Station roof through a
100 mm wide pipe. The target vented in this manner several times over the course of the
experiment (usually due to trouble with the refrigeration) and the safety systems worked
flawlessly. The scattering chamber is also equipped with a 160 mm wide pipe to vent the
hydrogen out the roof of ESA in the unlikely event of a rupture in the target loop itself.
This catastrophic failure mode never occurred.
42
3.7 Spectrometer
The E158 spectrometer is used to separate the Møller scattered electrons from backgrounds.
It is composed of a set of three dipole magnets followed by a package of four quadrupole
magnets. The spectrometer stretches roughly half of the length of End Station A, as can
be seen in Figure 3.19.
Figure 3.19: E158 spectrometer overview.
The design of the spectrometer is closely related to the geometry of the E158 calorimeter
(Section 4.1). Figure 3.20 depicts the face of the detector.
The calorimeter is divided into four annular regions designated In, Mid, Out, and eP.
The inner three regions are dominated by Møller scattered electrons and are used in the
primary measurement of APV . The outer region is dominated by electron-proton scatters
and is used for a supplementary measurement. All four regions are collectively called the
E158 calorimeter.
3.7.1 Dipole Chicane
The primary purpose of the dipole chicane is the collimation of background bremsstrahlung
photons. For the peak beam power of 500 kW, the target produces an 85 kW photon
43
Figure 3.20: Regions of the E158 calorimeter.
beam. The power level of the photon beam is too high to block with material. Instead, two
collimators are employed to keep the E158 calorimeter out of the line of sight of the target.
The unblocked photons travel with the unscattered primary beam electrons to the beam
dump out the east side of End Station A. Figure 3.21 depicts a 12 GeV scattered electron
and a target photon traversing the chicane.
Figure 3.21: E158 dipole chicane.
The photon collimation is provided by two tungsten cylinders located between the dipole
magnets. Figure 3.22 depicts the first of these collimators; the second is similar.
The chicane magnet strengths are chosen such that the integral of the transverse mag-
netic field over the path of a particle is zero, making it an achromat. Charged particles
emerge from the chicane along their initial trajectory, ensuring that the shape of the scat-
44
Figure 3.22: First photon collimator.
tered electron flux is unaffected by the presence of the magnets.
3.7.2 Main Acceptance Collimator
The main acceptance collimator is located between the end of the dipole chicane and the be-
ginning of the quadrupole package. The collimator acts in conjunction with the quadrupole
magnets to provide the separation of the Møller and eP scattered electrons. Figure 3.23
depicts the collimator, looking down the beamline. The flaring of the collimator edges is
done to minimize edge scattering. The collimator is 12 cm thick of copper, followed by 3
cm of tungsten, representing forty radiation lengths of material.
The primary beam and the signal flux of the luminosity monitor pass through the
central hole of the collimator. The Møller and electron-proton scatters observed with the
E158 calorimeter pass through the outer semi-circular holes. Figure 3.24 depicts the signal
flux at the calorimeter with and without the collimator in place. Note that the shape of the
distributions is greatly influenced by the quadrupole magnets, discussed in the following
section.
The In, Mid, and Out regions of the E158 calorimeter are located between radii of
45
Figure 3.23: Main acceptance collimator.
15 cm and 23.5 cm. With the collimator in place, these portions of the calorimeter are
dominated by the Møller scattered electrons between the energies of 12 GeV and 24 GeV.
The eP detector is positioned between 26.1 cm and 35 cm, where the signal is dominated
by electron-proton scatters.
3.7.3 Quadrupole Magnets
The quadrupole magnets shape the signal distribution that passes through the main ac-
ceptance collimator to separate the electron-electron and electron-proton scattering events.
Figure 3.25 presents data scans performed with the profile detector (Section 4.3). The re-
sults demonstrate the size of the signal flux at the E158 calorimeter with and without the
quadrupole magnets energized.
When the magnets are off, all of the electrons go to the eP detector. When the
quadrupoles are on, the lower energy Møller scattered electrons are focused onto the In, Mid,
and Out regions of the detector. Comparing quads-on and quads-off data with the Monte
Carlo simulation provides a powerful calibration of the model of the E158 spectrometer.
46
Figure 3.24: Effect of main acceptance collimator on signal flux.
Figure 3.25: Profile scans with and without quadrupole magnets.
The simulation (Section 6.7.1) is crucial for estimation of the background electron-proton
events in the In, Mid, and Out detectors.
3.7.4 Insertable Acceptance Collimator
An additional collimator is mounted on rails next to the main acceptance collimator. It
can be remotely inserted or withdrawn when required. It functions to decrease the size
of the acceptance of the main collimator, providing better separation between the Møller
47
and proton scattered electrons. The trade-off is that the amplitude of the overall signal
flux is greatly reduced. The collimator is inserted for applications that do not require high
statistics, such as polarimetry. Figure 3.26 is a photograph of the insertable collimator,
with the main acceptance collimator visible in the background.
Figure 3.26: Insertable acceptance collimator.
The large openings at the top and bottom of the insertable collimator are used for
polarimetry. The four smaller openings are used for background studies useful for fine-
tuning the Monte Carlo simulation.
Figure 3.27 depicts the change in the scattered electron profile with the insertable col-
limator in the beam. The left plot is a vertical scan, observing flux from the large lower
hole of the collimator. The separation between the Møller and proton scattered electrons is
cleaner than when the collimator is removed, shown in the plot on the right. The vertical
scale on the right plot has been normalized to match the conditions of the left plot, making
apparent the signal suppression with the insertable collimator in place.
3.7.5 eP Collimator
The asymmetry in the electron-proton flux that enters the eP detector is an order of mag-
nitude larger than APV . A simulation of showering in the E158 calorimeter demonstrates
48
Figure 3.27: Profile scans with (left) and without (right) the insertable collimator.
that shower leakage from the eP detector into the other regions of the detector can be an
important effect [43]. Shower leakage accounts for 35% of the eP background in the Out
detector, with roughly 15% in the Mid and In regions.
To reduce the influence of the background, the eP collimator was installed between Run
1 and Run 2 to block completely the eP detector, as well as 75% of the Out detector. The
collimator is composed of lead, 22.86 centimeters thick. It is a “clamshell” design, with
halves that are remotely insertable and removable. Figure 3.28 presents an overhead view
of the detector area, including the eP collimator, while Figure 3.29 depicts the coverage of
this collimator on the face of the E158 calorimeter.
Though the majority of the Out detector is blocked, its signal is only degraded by 25%
due to the eP collimator. The bulk of the Out signal resides at lower radii, and the Out
ring has some shower sharing with the Mid ring. Figure 3.30 presents data from profile
scans with and without the eP collimator in place. It is clear that the collimator effectively
blocks the flux of the eP detector.
49
Figure 3.28: Insertable eP collimator, top view.
3.7.6 Synchrotron Collimation and Photon Masks
A background of synchrotron photons is produced by the bends in the dipole chicane. For
the highest beam current and rate, the synchrotron radiation from the final dipole magnet
(D3) is 115 W, of which approximately 10% is directed at the E158 calorimeter [44]. The
energy in the synchrotron radiation intersecting the calorimeter is comparable to the energy
contained in the total electron signal flux. The synchrotron radiation represents a sizable
dilution to the main signal, but the primary concern is that it can introduce a helicity-
correlated asymmetry of its own, several orders of magnitude greater than APV [45].
The synchrotron background is decreased by three sets of collimators. The first is
provided by 40 radiation lengths of material in the spokes of the main acceptance collimator,
seen in Figure 3.23. The second is a similar pair of spokes, comprised of 20 radiation lengths
of tungsten, installed after the final quadrupole magnet. The final collimators are also spokes
of 20 radiation lengths of tungsten, bolted directly to the face of the detector. The final
two sets of synchrotron collimators are visible in Figure 3.31. The collimators reduce the
synchrotron signal to 0.15% of the electron signal in the E158 calorimeter.
50
Figure 3.29: The hatched area represents the coverage of the eP collimator on the face ofthe E158 calorimeter.
Figure 3.30: Profile scans without (left) and with (right) the eP collimator in place.
The final set of collimators are 7 tungsten rings located inside the drift pipe upstream
of the E158 calorimeter (Figure 3.31). These are used to block photons scattered from the
photon collimators between the dipole magnets. The rings were installed after the initial
engineering run of the experiment, before the beginning of Run 1. The resolution of the
Møller detector improved from 500 ppm to 200 ppm between Run 0 and Run 1, due to the
tungsten rings blocking this background.
51
Figure 3.31: Drift pipe synchrotron and photon collimators.
52
Chapter 4
Detectors
The data from the E158 experiment is used to determine a single quantity: the parity-
violating asymmetry in Møller scattering. However, the measurements of many different
detectors contribute both directly and indirectly to the final result. Figure 4.1 depicts the
locations of the detectors in End Station A. Most of the devices are mounted to a movable
cart, depicted in Figure 4.2. The following sections give descriptions of the E158 detectors
and their roles for the experiment1.
Figure 4.1: Overhead view of the detector locations in End Station A.
1The luminosity monitor was the primary responsibility of the author and is covered separately in Chap-ter 5.
53
Figure 4.2: Overhead view of the movable detector cart.
4.1 E158 Calorimeter
The E158 calorimeter is the primary detector of the E158 experiment [46]. It is divided
into four separate annular regions, denoted as the In ring, Mid ring, Out ring, and eP
detector. The spectrometer (Section 3.7) focuses electrons scattered from the target so that
the In, Mid, and Out regions are dominated by Møller scattering, while the eP detector is
dominated by electron-proton scatters.
The In and Mid region are collectively known as the Møller detector, and provide the
primary measurement of APV . The Out ring could also have been included, but Sec-
tion 6.7.2.2 will demonstrate that it was found to be susceptible to large systematic effects.
The eP detector supplements the Møller detector by providing information on the back-
ground electron-proton scattering events.
54
4.1.1 Calorimeter Design
The calorimeter is mounted on the movable detector cart located in End Station A (Fig-
ure 4.2). It is 16 radiation lengths thick, chosen as a compromise between large signal
size and minimizing sensitivity to the pion background. The calorimeter is composed of 100
copper plates interspersed with quartz fibers. The three inner regions are 10% quartz by vol-
ume, while the eP detector is 2%. Electrons shower in the copper and produce Cherenkov
light in the quartz, which is then directed through light guides to photomultiplier tubes
(PMTs) for detection. The Cherenkov angle in quartz is close to 45, so the plates and
fibers are positioned at this angle to maximize light collection efficiency. The resulting
geometry is depicted in Figure 4.3. The In, Mid, and Out rings make up the dark inner
region while the eP detector is the light outer region.
Figure 4.3: Partially constructed E158 calorimeter.
55
The quartz fibers are bundled to divide the calorimeter into the four concentric zones.
Each region is further subdivided with bundles of fibers servicing separate photomultiplier
tubes. Figure 4.4 depicts the different regions of the detector. The In and eP rings are
serviced by 10 tubes, while the Mid and Out rings have 20 tubes.
Figure 4.4: Calorimeter channel map.
To protect the PMTs from radiation damage, they are located roughly 70 cm from the
beamline. The 60 light guide periscopes that direct the Cherenkov light from the calorimeter
to the PMT locations are shown in Figure 4.5.
The PMTs are also encased by a large slab of lead for further protection. Figure 4.6
presents a diagram of the lead shielding. The PMTs are positioned in the cylinders drilled
in the back shield. The two rows of PMTs are located at radial distances of 67.3 cm and
75 cm. Over the course of the experiment, the E158 calorimeter absorbed a radiation dose
56
Figure 4.5: Light guide configuration.
of approximately 500 MRad, while the PMT dose was only ∼1 Rad.
Figure 4.6: E158 calorimeter lead shielding.
4.1.2 Calorimeter Electronics
The electronics for the E158 calorimeter are depicted in Figure 4.7 [47]. The RLC circuit
is employed to increase the length of the signal from the PMTs. The longer time constant
allows the ADCs (analog-to-digital converters) to integrate for a longer period of time,
57
suppressing random noise. Employing this method, it was found that the ADCs had only
one to two counts of noise, compared to the full range of 64,000 counts.
Figure 4.7: E158 calorimeter electronics diagram.
The power supplies and ADCs are located in the electronics hut, which is accessible at
all times. The hut is connected to the End Station through 200 feet of cable laid in an
underground tunnel. The amplifiers are located in the End Station near the detector to
avoid amplifying pick-up noise from the cables.
4.1.3 Calorimeter Resolution
The asymmetry resolution of the Møller detector is the dominant contributor to the overall
uncertainty on the measured value of APV . Because of the large cross section for Møller
scattering, the detector receives a signal of ≈ 20 million scattered electrons for a beam cur-
rent of 5×1011 electrons. The counting statistics contribute roughly 160 parts-per-million
(ppm) to the resolution of the detector. Additionally, common-mode electronics noise con-
tributes 110 ppm to the resolution, so that the overall resolution is near 200 ppm. Figure 4.8
depicts the Møller detector asymmetry distribution for a standard one-hour data run. The
correlation of the detector with the beam monitors has already been removed through the
regression process, covered in Section 6.3.
58
Figure 4.8: Møller detector asymmetry resolution.
The Out detector receives an additional four to seven million Møller electrons (depending
on whether the insertable eP collimator is in or not). If the Out detector is included with
the In and Mid regions, the detector resolution improves slightly, as seen in Figure 4.9.
Because of its large systematic susceptibility and only marginal resolution gain, the Out
detector is not included in the measurement of APV .
Figure 4.9: Møller plus Out ring asymmetry resolution.
The eP ring is dominated by electron-proton scatters, and is used to measure the parity-
59
violating asymmetry in this background. Figure 4.10 depicts the eP detector asymmetry
distribution. The resolution of the eP detector is adequate to perform its function, which
will be covered in Section 6.7.1.
Figure 4.10: eP ring asymmetry resolution.
4.1.4 Calorimeter Linearity
A non-linear response can introduce an error between the asymmetry observed with the
detector AMeasured and the physics asymmetry APV . Section 5.10.1 demonstrates that the
shift introduced by the non-linearity ε of the detector is given by
AMeasured = (1 − ε)APV − εAToroid, (4.1)
where AToroid refers to the charge asymmetry of the beam measured with a toroid.
The regression procedure is employed to reduce the detector’s sensitivity to beam pa-
rameters, including the charge asymmetry, effectively removing the εAToroid term in Equa-
tion 4.1. The remaining (1 − ε)APV term is a source of systematic uncertainty. Because
APV is ≈ -150 ppb, it is important to keep ε at the level of 1% to insure that the systematic
60
Run IRing LinearityIn 0.996 ± 0.013
Mid 0.986 ± 0.010Out 0.995 ± 0.012
Run IIRing LinearityIn 0.994 ± 0.012
Mid 0.989 ± 0.009Out 1.009 ± 0.011
Table 4.1: Measured linearity of the In, Mid, and Out rings of the E158 calorimeter, forRun I and Run II.
uncertainty is only a few ppb.
The linearity of the calorimeter was constrained by comparing the observed value of
a large asymmetry measured at multiple PMT input light levels [48]. The asymmetry
was provided by the iron foil used for polarimetry, described in Section 4.2. The foil was
inserted simultaneously with the liquid hydrogen target so that the signal flux was nearly
the same as during normal production running. Filters were placed in front of some of the
PMTs, reducing the light level to 1/3 or 1/2 of the signal seen during normal running. By
comparing the asymmetry recorded by each class of PMT, it is possible to determine the
linearity. Linearity of 100% would mean that all tubes would record the same asymmetry.
Table 4.1 presents the Møller detector linearity found by this method in Run I and Run II.
The results indicate that the detector non-linearity is at the 1% level, as required.
4.2 Polarimeter
The polarimeter is used periodically to monitor the polarization of the electron beam.
Measurements are performed after each energy change and source halfwave plate toggle,
corresponding roughly to one measurement every two days. In the absence of other back-
grounds, the physics asymmetry APV is related to the asymmetry measured with the Møller
61
detector AMeasured and the beam polarization P through
APV =AMeasured
P. (4.2)
The uncertainty on P translates directly into an uncertainty on APV .
4.2.1 Polarimetry Method
The beam polarization measurement is performed in a special configuration of the E158
apparatus, with the liquid hydrogen target retracted and a supermendur foil placed in
the beam [49]. The foil is magnetically polarized by a pair of Helmholtz coils which are
only energized during the measurement. The insertable acceptance collimator, discussed in
Section 3.7.4, is moved in, and the spectrometer quadrupole magnets are adjusted to provide
better separation of the Møller and electron-proton scattering events. The polarimeter
detector, described in the following section, is moved onto the Møller peak in the scattered
flux, directly in front of the Møller detector.
The polarized foil produces a helicity-correlated asymmetry in the scattering rate, due to
the magnetic dipole interaction [50]. The asymmetry Am is related to the beam polarization
Pbeam, the target polarization Pfoil, and the center-of-momentum scattering angle θCM
through
Am = cos(20)PbeamPfoil(7 + cos2 θCM ) sin2 θCM
(3 + cos2 θCM)2. (4.3)
The coefficient of cos(20) is included to account for the angle of the foil with respect to
the beam. The tree-level asymmetry calculation of Equation 4.3 is adequate because higher
order corrections are found to be well below 1% [51].
The foil is comprised of 49% Cobalt, 49% Iron, and 2% Vanadium. All three of these
62
elements have two valence electrons that are polarized by the magnetic field, while the inner
electrons remain unpolarized. The valence electrons account for roughly 8% of all electrons,
so the maximum polarization of the foil is around 8%. Assuming a beam polarization of
75%, Am is expected to be approximately 4%, over five orders of magnitude larger than
APV .
Because the asymmetry is large, it can be measured in a relatively short period of
time. To avoid excessive heating of the foil, runs are taken at a beam repetition rate of
10 to 15 Hz, compared to the normal rate of 120 Hz. For each measurement, data is
taken for approximately 10 minutes, accumulating 5000 pulse pairs. Figure 4.11 presents
the measured asymmetry distribution for a typical polarimetry run. While the statistical
Figure 4.11: Polarimeter asymmetry distribution for a single run.
uncertainty on the measured asymmetry is small, the polarimetry systematic uncertainties
covered in Section 4.2.3 dominate.
Immediately after each polarization run, data is taken with the foil removed, to serve as
a background measurement. In total, each polarization measurement lasted approximately
30 minutes.
63
4.2.2 Polarimeter Design
The polarimeter is mounted on the detector cart between the profile monitor and the Møller
detector, as shown in Figure 4.2. Figure 4.12 is a schematic of the detector looking down-
stream from the target.
Figure 4.12: Polarimeter position on the detector cart.
The apparatus is remotely movable in the vertical direction. When not in use, the
polarimeter is lowered to a radial distance of 50 cm so that it does not interfere with the
electron flux entering the Møller detector. During a polarimetry measurement, the device
is raised to a radial distance of 21 cm, the location with the highest rate of scattered Møller
electrons.
Signal electrons create an electromagnetic shower in the polarimeter’s quartz-tungsten
sandwich, which is depicted in Figure 4.13 [52]. The shower creates Cherenkov light which
is directed through a light guide to the photomultiplier tube that records the signal. The
sandwich plates are angled at 30, which simulations indicated was optimal for light collec-
tion. The extra tungsten plate at the front of the sandwich has only a minimal influence
64
Systematic UncertaintiesItem Uncertainty
Background Subtraction 3.0%Foil Polarization 3.0%Result Spread 2.5%
Polarimeter Linearity 1.5%Foil Angle 1.0%
Foil Heating Depolarization 1.0%Levchuk Effect [54] 1.0%Analyzing Power 0.5%
Helmholtz Magnetic Field 0.2%
Table 4.2: Polarimetry uncertainties, relative to Pbeam.
on the light yield, but serves to decrease sensitivity to low-energy backgrounds.
Figure 4.13: Quartz-tungsten sandwich of polarimeter.
4.2.3 Polarimeter Systematic Uncertainties
While the data acquired for each polarization measurement provides a statistical uncertainty
well below 1%, there are many sources of systematic uncertainties also to consider. Table 4.2
itemizes each effect, which are covered in detail in Reference [53].
The contribution labeled “Result Spread” refers to the results obtained with different
target foils. Three foils of differing thickness were available on the movable target chassis:
20 µm, 50 µm, and 100 µm. Each foil gave a different result for the polarization. The reason
65
for this behavior is unclear, so the spread in the results is submitted as a contribution to
the systematic uncertainty.
The quadrature sum of all the items nets a total uncertainty of 5.5%, relative to Pbeam.
The beam polarization was typically around 84%, so this amounts to an absolute uncertainty
of ±4.6% on the beam polarization.
4.2.4 Polarimeter Results
Figure 4.14 shows the polarization measurements spanning one month of Run II. The vari-
ation among the measurements is only a few percent, indicating that the beam polarization
was fairly stable. The average polarizations used for computing the physics asymmetry
were found to be
Pbeam = 84.9% ± 4.6% (4.4)
for Run I and
Pbeam = 84.4% ± 4.6% (4.5)
for Run II.
4.3 Profile Monitor
The profile monitor is used to measure the radial distribution of the scattered electrons en-
tering the Møller detector. The signal flux maps are employed to fine-tune the Monte Carlo
simulation of the E158 spectrometer, which plays a critical role in background estimation
for the Møller detector measurements.
66
Figure 4.14: Run II beam polarization measurements, spanning 1 month.
4.3.1 Profile Monitor Design
Figure 4.2 shows the location of the profile monitor from overhead, while Figure 4.15 presents
a schematic drawing of the device. The profile monitor is comprised of two of pairs movable
Cherenkov detectors mounted to a rotatable annulus. The detectors can be moved radially
from the beam pipe out to 50 cm, fully covering the extent of the E158 calorimeter. The
annulus can be rotated over 180, allowing radial scans at any angle. The scanners are
cross-calibrated by comparing data taken 180 apart.
The Cherenkov detectors are mounted in pairs for background subtraction [55]. The
signal from the front scanner is weighted and subtracted from the back scanner to produce
a corrected signal. The front scanner utilizes only a quartz block, while the back scanner is
also equipped with a tungsten preradiator. Figure 4.16 depicts one of the back scanners.
The shutter mounted in front of the PMT can be opened and closed remotely, for
background studies. Additionally, the outer tube of the scanner is rotatable so that the
tungsten preradiator can be removed for additional background measurements.
67
Figure 4.15: Profile monitor schematic.
4.3.2 Profile Scan Results
A profile scan consists of multiple radial scans over the full azimuth, in increments of 20. A
full scan required roughly 30 minutes of beam time. Figure 4.17 shows a typical radial scan
at one particular angle. The Møller peak is at the left, and the electron-proton scattering
peak is at the right. The use of profile scans for background subtraction is discussed in
Section 6.7.1.
4.4 Pion Detector
Pion production in the target contributes a small background to the Møller scattering
process. The parity-violating asymmetry APV is related to the asymmetry obtained from
68
Figure 4.16: Quartz scanner schematic.
Figure 4.17: Radial scan produced with the profile monitor.
the Møller detector AMeasured and the pion background through
APV =AMeasured − εNπ
NeeAπ
1 − εNπNee
Aπ. (4.6)
The term Nπ/Nee is the ratio of the number of pions to electrons passing through the
Møller detector. The ε term refers to the average signal size produced by a pion in the
Møller detector, compared to the average signal produced by an electron. The term Aπ
represents the parity-violating asymmetry of the pion background. The pion detector is
utilized to determine Aπ, as well as the ratio Nπ/Nee, while the remaining unknown ε must
69
be found from simulations of the Møller detector [56].
Pions are produced through real and virtual photoproduction, as well as deep inelastic
scattering processes. Calculations indicate that the pion background passing through the
Møller detector is likely to have an asymmetry at the level of 1 ppm, an order of magnitude
higher than the Møller scattering asymmetry APV [37]. However, it is also found that
the ratio Nπ/Nee is expected to be less than 1%. Sensitivity to the background is further
suppressed because the Møller detector is less sensitive to pions than electrons. Simulations
show that ε is 0.22 ± 0.15. Combining these factors, it is expected that the pion asymmetry
correction to AMeasured is less than 4 ppb, with a dilution factor less than 0.004.
4.4.1 Pion Detector Design
The pion detector is located directly behind the E158 calorimeter, shown in Figure 4.2. The
lead PMT shield and active copper region of the Møller detector constitute 60 radiation
lengths of material in front of the pion detector. The material ensures that the signal in the
pion detector is dominated by pions, with little contamination from scattered electrons.
The detector is composed of 10 individual photomultiplier tubes, mounted symmetrically
around the beam pipe. A quartz block is positioned in front of each tube, defining an active
region for the detector between 15 cm and 23.5 cm, measured radially from beam center.
This region exactly matches the extent of the three inner rings of the E158 calorimeter so
that the two detectors observe the same pion flux. The tubes are mounted at 45 relative
to the beam, to maximize Cherenkov light collection efficiency. Figure 4.18 depicts the pion
detector looking downstream from the target.
70
Figure 4.18: Pion detector layout.
4.4.2 Pion Detector Resolution
The required asymmetry resolution for the pion detector is set by the general goal that indi-
vidual systematic uncertainties be kept below the level of 5 ppb. Because the correction due
to the background pion asymmetry is expected to be less than 5 ppb for the Møller detector,
it is acceptable to measure the pion asymmetry to ± 100% of its value, corresponding to
a precision of ≈ 1 ppm. Since Run I and Run II are each composed of roughly 80 million
pulse-pairs of data, the RMS of the pion detector asymmetry distribution is required to be
less than 1 ppm×√8 × 107 ≈ 0.01.
Figure 4.19 presents the pion asymmetry distribution for a typical one-hour data run.
The RMS of the distribution is a factor of two smaller than the required 0.01, indicating
that the pion detector design is adequate for the experiment.
It should be noted that the resolution is 25 times less sensitive than the Møller detector.
Because of its relative insensitivity, it was found that it was unnecessary to use the regression
procedure to remove the correlation of the pion detector asymmetry to beam asymmetries:
all of regression correlation coefficients were consistent with zero.
71
Figure 4.19: Pion detector asymmetry distribution, covering one run.
4.4.3 Pion Detector Electromagnetic Background
Though the pion detector is positioned behind 60 radiation lengths of material, it is still
expected that Møller scattered electrons will contribute a small dilution to the pion signal.
To measure the amount of contamination, special runs were taken with the polarimetry
foil included with the liquid hydrogen target. The runs have a large, quickly measured,
asymmetry. For these data, the Møller detector recorded an asymmetry of 522 ppm (with
negligible uncertainty) while the pion detector measured an asymmetry of 62 ppm ± 11
ppm. Therefore, it is demonstrated that the electromagnetic contamination in the pion
detector is only 13.5% ± 2.7%, which is small. The remaining signal in the pion detector is
assumed to be pions.
4.4.4 Ratio of Nπ to Nee
The ratio Nπ/Nee was determined from two special runs. First, one of the pion detector
tubes was placed in front of the Møller detector. Data was then taken using only the
polarimetry foil as a target. The foil was unpolarized: it was used simply as a thin target,
72
so that the pion tube would not saturate. The tube was then moved back behind the
Møller detector, and normal data was taken with the liquid hydrogen target. The signal
attenuation provided by the Møller detector is related by
fAttenuation =PMTBackLH2
PMTFrontFoil
× MøllerFoil
MøllerLH2, (4.7)
where the term fAttenuation designates the ratio of the signal flux at the front of the Møller
detector to the signal at the front of the pion detector. The PMT terms refer to the
signal size from the pion tube, while the Møller terms designate the signal size found by
combining all of the tubes from the Mid ring of the Møller detector. The Møller ratio is
included as a normalization, to account for the use of the different targets. It was found
that fAttenuation = 2.003 × 10−4 ± 0.346 × 10−4.
The attenuation number is compared with a GEANT3 simulation to extract the ratio
Nπ/Nee [57]. The simulation is used to determine the value of
fAttenuation =CBackee + Nπ
NeeCBackπ
CFrontee + NπNee
CFrontπ
. (4.8)
Here the C terms refer to the size of the Cherenkov response of the pion detector. The
Back and Front labels indicate the location of the pion tube, while ee and π refer to Møller
electrons and pions, respectively.
Comparing the observed fAttenuation with the simulation yields
Nπ
Nee= 0.0063 ± 0.0021. (4.9)
The uncertainty is composed of the statistics of the real measurement, as well as a conser-
73
Pion Asymmetry AπRun I -1.74 ± 0.46 ppmRun II -0.36 ± 0.48 ppm
Table 4.3: Pion detector asymmetry results.
vative systematic uncertainty estimate found by varying the simulation parameters.
4.4.5 Pion Detector Results
Table 4.3 presents the asymmetries measured with the pion detector for Run I and Run II.
The results for the two runs differ by three standard deviations. The cause for this effect
can be understood by noting that the signal in the pion detector is reduced by 40% when
the insertable eP collimator (Section 3.7.5) is in place. This indicates a sizable signal due to
punchthrough from the eP detector directly into the PMTs of the pion detector. Because
the asymmetry measured with the eP detector was ≈ 1.5 ppm, the contamination of the
pion detector in Run I renders it unusable. Therefore, it was decided that the Run II result
would also be employed for Run I, with the Run I result disregarded. Using Aπ, Nπ/Nee,
and ε, the correction applied to Run I and Run II is -0.5 ± 0.8 ppb, with a dilution factor
εNπNee
of 0.0014 ± 0.0011.
4.5 Synchrotron Light Monitor
The Synchrotron Light Monitor (SLM) detects synchrotron radiation produced in the A-
Line, the curved beamline connecting the linac to End Station A [58]. The general location
of the detector is depicted in Figure 4.20.
The device is used to quantify the vertical polarization of the beam by measuring the
helicity-correlated asymmetry in the synchrotron radiation. The asymmetry has relevance
to the Møller detector, because it also presents itself in the synchrotron radiation background
74
Figure 4.20: SLM location in the A-Line bend.
produced in the dipole magnets in End Station A.
4.5.1 Synchrotron Asymmetry
The power emitted by an electron beam traversing a magnet has an asymmetry related to
the amount of spin polarization in the direction of the magnetic field [45, 59]. The power
in a synchrotron beam is given by
P = P0
[1 −
(55√
324
+ η
)χ
], (4.10)
where P0 refers to the classically calculated synchrotron radiation power, and η = ±1 refers
to the electron spin oriented parallel or anti-parallel to the magnetic field. The χ term
reflects the properties of the applied magnetic field through
χ =3hγ2
2mecρ, (4.11)
with ρ being the bend radius. The asymmetry in the synchrotron radiation power is found
to equal χ through
ASR ≡ P+ − P−
P+ − P− = χ. (4.12)
The final dipole magnet in the E158 spectrometer, D3, produced the vast majority of
75
the synchrotron radiation background for the Møller detector. Using the values for this
magnet, it is found that the power asymmetry in the synchrotron radiation is ≈ 35 ppm
for a 100% vertically polarized electron beam [44].
If the Møller detector were truly a calorimeter for photons over the energy range in the
synchrotron radiation, the analyzing power for vertical polarization would be χ. However, it
is likely that the detector has a non-linear response to synchrotron photons in the relevant
energy ranges. Simulations indicate that the energy weighted asymmetry for the Møller
detector would be closer to 65 ppm, rather than the real asymmetry of 35 ppm [60]. To be
conservative, it will be assumed that the Møller detector has an analyzing power of 60 ppm
± 30 ppm for vertical beam polarization.
By comparing the Møller detector signal sizes for empty target runs to normal runs
with the liquid hydrogen target, it was found that synchrotron radiation comprises 0.15%
± 0.05% of the total Møller detector signal. Combining this factor with the synchrotron
asymmetry and an expected vertical beam polarization of ∼1% reduces the effect of the
background to a few ppb.
The asymmetry measured with the SLM is used to place bounds on the amount of verti-
cal polarization in the electron beam, which in turn constrains the effect of the synchrotron
radiation produced in the D3 magnet on the Møller detector.
4.5.2 SLM Design
Figure 4.21 presents a schematic of the SLM. The synchrotron radiation in the SLM region
exits through a 1 cm thick aluminum flange. Then, a 1 mm thick layer of lead is used
to convert the photons into an electron shower. The shower then traverses a quartz bar,
producing Cherenkov light. The light is directed by a mirror to three photodiodes, which
76
deliver the SLM signal to the ADCs. Photodiodes have low resistance to radiation, so the
mirrored box is heavily shielded with lead.
Figure 4.21: SLM layout.
4.5.3 SLM Resolution
The synchrotron radiation background affects the Møller asymmetry in a manner analogous
to the pion background given in Equation 4.6:
APV =AMeasured − fASynch
1 − f. (4.13)
The term f is the dilution factor, while ASynch refers to the asymmetry registered in the
Møller detector due to the synchrotron radiation produced in the D3 magnet of the spec-
trometer.
The value for fASynch is given by
fASynch = (0.0015 ± 0.0005) × (60 ppm ± 30 ppm) × Py, (4.14)
where the first term is the dilution factor f , the second is the analyzing power of the
77
Møller detector, and Py is the vertical beam polarization. As with the pion background,
it is desirable to keep the uncertainty due to the synchrotron background below 5 ppb.
Therefore, the SLM is required to measure the vertical beam polarization to roughly 5%.
The analyzing power of the SLM is found from simulations to be 60 ppm ± 30 ppm
(coincidentally the same as the Møller detector). Therefore, over the course of the typical
data set of 80 million pulse pairs, the SLM must measure its asymmetry to the level of 2
ppm, corresponding to an asymmetry resolution of 0.03.
It was found that combining the three SLM channels actually did not significantly in-
crease the detector resolution. The dominant noise source for the detector must be common
to all three photodiodes. Therefore, it was decided to use the single SLM channel (diode 2)
with the best resolution for the SLM result. Figure 4.22 depicts the asymmetry distribution
of the SLM data for a typical one-hour data run.
Figure 4.22: Regressed SLM channel 2 asymmetry distribution, for one run.
With an RMS of 840 ppm, the SLM greatly exceeds the resolution requirements of the
E158 experiment. Unlike the pion detector, the resolution is sufficient that regression against
beam monitors is useful. Like the Møller detector and the luminosity monitor, the SLM is
78
Vertical Polarization (%)45 GeV 48 GeV
Run I -1.7 ± 0.9 1.7 ± 0.9Run II -1.7 ± 0.9 3.1 ± 1.6
Table 4.4: Vertical beam polarization at the target.
regressed against beam charge, energy, X and Y position, and X and Y angle. Regression
produces a factor of two improvement in the resolution, resulting in the distribution seen
in Figure 4.22.
4.5.4 SLM Results
The SLM asymmetry results are presented in Figure 4.23. The data is split into four separate
chunks, based on energy and source halfwave plate (HW) state. The cancellation between
the energy states is good, effectively reducing the systematic effect due to synchrotron
radiation for the Møller detector.
Figure 4.23: SLM asymmetry results for Run I and Run II.
Table 4.4 presents vertical beam polariation results for both Run I and Run II, computed
using the SLM asymmetry. The amount of vertical polarization is ∼1%, consistent with
expectations.
Using the Møller detector analyzing power and the synchrotron dilution factor, the
79
Diluted Synchrotron Asymmetry (ppb)45 GeV 48 GeV
Run I -1.6 ± 1.2 1.5 ± 1.2Run II -1.6 ± 1.2 2.8 ± 2.2
Table 4.5: Synchrotron asymmetry correction for the Møller detector.
diluted asymmetry contribution to the Møller detector asymmetry fA can be calculated.
Table 4.5 details the results, which are found to be at a manageable level.
80
Chapter 5
Luminosity Monitor
The luminosity monitor is designed to perform measurements complementary to the Møller
detector. The primary function of the device is to provide a statistically significant null-
asymmetry measurement by observing extremely forward angle Møller and Mott scattering.
It fills the secondary role of determining the level of density fluctuations in the liquid
hydrogen target. The detector is also used to monitor the noise properties of the beam
because of its large sensitivity to beam motion. All of these functions serve to ensure the
data quality that is used in producing the final physics result.
5.1 Detector Signal
The luminosity monitor, henceforth to be called simply the “lumi,” is positioned 70 meters
downstream of the target, at an angle of one milliradian (Figure 5.1). At this location,
about 70% of the scattered electrons that hit the detector have energies greater than 40
GeV. The average Q2 of the signal is 0.003 (GeV/c)2, an order of magnitude lower than for
the signal observed with the Møller detector [61].
The signal rate in the detector is high, with approximately 3×108 scattered electrons for
the nominal beam current of 5×1011 electrons per pulse. It is because of the large signal in
this region that the lumi is able to provide an asymmetry measurement that has statistical
81
Figure 5.1: Layout of End Station A for experiment E158.
significance on the level of the result obtained by the Møller detector. Figure 5.2 depicts a
Monte Carlo simulation of the various components of the lumi signal flux [62].
Figure 5.2: Components of lumi signal at face of detector.
The signal is dominated by Møller and Mott scattering from the target. The contribution
labeled eA refers to scattering from components in the experimental apparatus other than
the target. The largest contributor to this background is the aluminum end windows of
the target cell. The final component is a small contribution from inelastic electron-proton
82
scattering in the target.
The overall expected asymmetry Alumi is the sum of the individual signal component
asymmetries Ai, weighted by their signal size Si through
Alumi =∑i SiAi∑i Si
. (5.1)
Figure 5.3 depicts the asymmetries of each of the lumi signal components, multiplied by the
ratio of its signal size to the total signal. (Note that the sign of the eA asymmetry has been
reversed.) Combining each component yields an expected asymmetry of -15 ppb ± 5 ppb
Figure 5.3: Contributions to lumi asymmetry.
for the lumi. For comparison, the Møller detector is expected to observe an asymmetry of
approximately -150 ppb, a full order of magnitude larger.
Over the course of the experiment, the Møller detector is expected to observe a non-zero
asymmetry with a significance of 10 standard deviations. The lumi is expected to measure
83
essentially zero asymmetry, with the same statistical significance. As a result, the lumi can
be used to augment the Møller detector as a sensitive monitor for false asymmetries.
5.2 Synchrotron Radiation Background
The E158 spectrometer, which is discussed in Section 3.7, has three dipole magnets used to
direct the primary beam around the collimation for photons from the target. The deflected
beam produces synchrotron radiation, which in principle could be a large background for the
lumi. The amount of power in the synchrotron beam is calculated to be about 0.02% of the
total beam power [44]. For nominal data runs, this amounts to approximately 115 Watts.
Simulations show that about 15% of this power is directed at the luminosity monitor [63].
Figure 5.4 displays a simulation of the distribution of background photons at the face of
the lumi. The dark horizontal band is the region dominated by synchrotron radiation. The
density of photons is weighted predominantly to the right side of the plot because the bend
of the final dipole is to the left. The sensitive region of the detector is superimposed as the
area between the black circles. The empty central region was omitted from the simulation
for simplicity.
The energy absorbed from actual signal electrons is expected to be about 150 Watts,
meaning that synchrotron radiation is roughly 10% of the energy of the lumi signal at the
face of the detector [64]. By itself this is a large dilution factor, but the main issue of concern
is that synchrotron radiation can have helicity-correlated asymmetries of its own [45, 59, 65,
66]. Asymmetries up to the level of 600 ppb could be generated in the synchrotron spectrum
from the E158 spectrometer. Because the expected physics asymmetry of the lumi is only
15 ppb, it is desirable to limit the detector sensitivity to synchrotron radiation to less
than 1% of the total signal. As will be demonstrated in Section 5.9, this was successfully
84
Figure 5.4: GEANT simulation of photon flux at the lumi.
incorporated into the design and renders the synchrotron background contribution small.
5.3 Detector Design
The lumi is an ionization detector, comprised of 16 individual channels. These channels are
grouped into two separate, full-azimuth rings. The upstream ring is identified as the “front
lumi” and the downstream ring as the “back lumi.” The front lumi is positioned behind
seven radiation lengths of aluminum showering material. Directly behind the front lumi is
an additional four radiation lengths of aluminum, followed by the back lumi. Figure 5.5
depicts the layout of the full luminosity monitor.
The aluminum between the two sets of rings ensures that the back lumi will have a
smaller signal than the front lumi. An EGS4 simulation indicates that the reduction in
signal should be approximately a factor of three between the two rings [64]. The smaller
signal allows the back lumi to serve as a cross-check on the front ring for systematic effects
85
Figure 5.5: A. Front view of one full lumi ring, with sensitivity between 7 and 10 cm. B.Side view, depicting the two lumi rings and the aluminum showering material.
related to signal size.
Each chamber houses a package of 11 parallel plates (Figure 5.6), positioned transverse
to the incoming signal electrons. A bias of 100 V is applied to alternate plates, in order
to produce an electric field between each plate pair. As an ionizing particle traverses the
chamber, it produces electron-ion pairs, which are collected on the plates as the signal.
Figure 5.6: Individual chamber design, with signal plates shaded.
The chambers are filled with nitrogen gas, at a pressure of just slightly over one at-
mosphere. The overfilling is done to ensure that the amount of oxygen in the chambers
is minimized, since it has the property that it can capture signal electrons before they
are collected on the plates. The gas is continually flowed through the chambers so that
86
contaminants due to radiation damage are not allowed to accumulate.
An electron passing through the lumi parallel to the beamline encounters one centimeter
of nitrogen. It is expected that a single electron will produce ≈ 60 ionization pairs for this
amount of gas [67].
The aluminum in front of the detector rings performs the dual role of showering the main
signal and attenuating synchrotron radiation. A simulation shows that the front lumi ring
is positioned very close to the shower maximum, with a predicted multiplicity of roughly
100 [64]. It is also expected that the synchrotron radiation background is suppressed below
the level of 1% of the signal at this depth.
The large scattering rate, coupled with the gain factors from the showering in the alu-
minum and the ionization trail in the nitrogen gas, allows the chambers themselves to
produce signals on the level of several volts. The size of the output signal means that no
additional amplification electronics are required, which considerably simplifies the detector
design.
The signals travel along 100 feet of coaxial cables from the chambers to the ADCs, where
they are read in differentially (after the bias voltage is filtered out of the signal cable). By
subtracting the two signals from each chamber, noise pick-up from the transmission cables
is suppressed. Figure 5.7 is a schematic of the lumi electronics setup.
5.4 Lumi Signals
Because each pair of plates is capacitively coupled, each chamber actually produces two
signals, of opposite sign. Figures 5.8 and 5.9 display oscilloscope traces of these signals, the
first from a front lumi chamber and the second from a back lumi chamber.
The time structure of the beam provided by the accelerator can be approximated as a
87
Figure 5.7: Lumi electronics setup.
Figure 5.8: Front lumi signal traces.
square pulse with a duration of 300 ns. As can be seen from the scope traces in the figures,
the lumi signal is of similar length, indicating that electron collection occurs quickly. Also
note that the front lumi has a signal three times larger in magnitude than the back lumi,
in agreement with simulations.
The fast signal observed on the scope is composed of the electrons from ionization pairs.
The ions arrive much later than their electron counterparts due to their much higher mass.
In fact, the ion signal is stretched out to such an extent that it is all but invisible on
the oscilloscope traces. Nevertheless, it is statistically beneficial to collect the signal due
88
Figure 5.9: Back lumi signal traces.
to the ions. By extending the integration time of the ADCs from two microseconds to
30 microseconds, thereby collecting more ions, it is observed that the detector asymmetry
resolution improves by 50%.
5.5 Gas System
The gas system (Figure 5.10) for the lumi was designed to be very simple. The nitrogen
for the entire detector is supplied by a single bottle, located outside of End Station A. The
flow rates of the individual chambers are balanced by adjusting needle valves located on the
the output lines. During operation, the gas flow is monitored by microphones positioned
to detect the sound of nitrogen bubbles escaping through silocone oil at the end of the
gas lines. The flow rate was extremely low, and the gas bottle was changed only once per
month. The chambers are electrically isolated from each other by inserting non-conductive
tubing at the entrance and exit ports of the copper gas lines.
89
Figure 5.10: Gas system configuration for lumi.
5.6 Asymmetry Resolution
The asymmetry measured with the lumi ARaw is defined as
ARaw =
(ST
)R −(ST
)L(ST
)R+(ST
)L , (5.2)
where S refers to the detector signal, L and R are the beam helicity, and T refers to the
beam charge measured by a toroid just upstream of the target.
To measure the true physics asymmetry, however, effects due to beam motion must
be removed. This is achieved through measuring the correlation between the detector
asymmetry and the various beam monitors, and then removing this correlation. This process
is known as beam regression, and will be discussed in detail in the section concerning data
analysis (Section 6.3). Regression covers six parameters, removing the detector correlations
with the X and Y beam positions, dX and dY beam angles, the beam energy E, and the
beam charge Q.
90
The asymmetry is corrected on a pulse-pair basis through
APV = A−6∑
n=1
cn(bRn − bLn), (5.3)
where the sum covers the six regression parameters, the constants cn are the experimentally
determined correlation coefficients, and the bn terms are the values given by the BPMs. In
the case of the charge correction, the difference in Equation 5.3 becomes an asymmetry
measured with a toroid.
Figures 5.11 and 5.12 display the correlation coefficients for the position and angle
motion obtained from a typical run, for each lumi chamber. The large azimuthal dependence
of the coefficients is evident in the sinusoidal pattern in the plots. The numbering scheme
for the chambers is the same as found in Figure 5.5.
Figure 5.11: X and Y correlation coefficients. The averages are shown as straight lines.
The asymmetry for the whole detector is the average of the asymmetries found by the
individual chambers, given by
APVTotal =1N
N∑m=1
APVm . (5.4)
91
Figure 5.12: dX and dY correlation coefficients The averages are shown as straight lines.
Here the sum is over the individual lumi channels, and N refers to the total number of
channels used. Since the front lumi and back lumi are treated as separate detectors, this
number is usually eight.
Substituting Equation 5.3 into Equation 5.4 for each chamber yields the expression for
the whole detector through
APVTotal =1N
N∑m=1
(ARawm −6∑
n=1
cmn (bRn − bLn)), (5.5)
which can be written as
APVTotal =< ARaw > −6∑
n=1
< cn > (bRn − bLn). (5.6)
The variable m is summed over the lumi channels, and n sums over the beam monitors.
The form of the asymmetry given in Equation 5.6 is exactly the same as in Equation 5.3,
except that the terms are replaced by averages over the chambers. In particular, it can be
seen that the correlation coefficient relevant for the detector as a whole is the average of
92
the individual chamber correlation coefficients.
The average of the channel coefficients is plotted as a solid line in Figures 5.11 and 5.12.
The average is always much less than the amplitude seen in the coefficients of the individual
chambers, indicating that the sensitivity of the detector as a whole to beam motions is
much less than the individual chamber sensitivities. The suppression is so large because
lumi chambers are automatically gain matched by design.
The RMS of the distribution of APV is the resolution of the detector. For a given set
of data, the resolution determines how well the mean asymmetry is known. The Møller
detector typically has an RMS of 200 ppm. For the lumi to serve as a useful cross-check
on the main detector, then, it must also have an asymmetry resolution of at least 200
ppm. Figure 5.13 demonstrates the level of improvement in resolution obtained by removing
detector correlations to the beam through regression. The RMS of the corrected distribution
is roughly 100 ppm, which is at a level that meets experimental goals for the lumi design.
Figure 5.13: Left: Raw lumi asymmetry distribution. Right: Regression-corrected lumiasymmetry distribution.
93
5.7 Resolution Contributions
In order to evaluate the performance of the lumi, it is instructive to itemize the contributions
to the overall asymmetry resolution. Mathematically, it can be shown that the expected
resolution is given by
σ2APV =
12N
(1+∆EE
2
)+σ2Boiling+σ
2Pedestal+(1+cQ)2σ2
ToroidResolution+5∑
n=1
c2nσ2BPMResolution.
(5.7)
The term σAPV refers to the statistical width of the asymmetry distribution. The number
of scattered electrons entering the detector is given by N , so the 12N term is a reflection
of counting statistics. The statistics are modified by the energy resolution of the detector,
represented by the ∆EE term. The next term, σBoiling, refers to contribution due to density
fluctuations in the liquid hydrogen target. The term σPedestal represents all types of elec-
tronics noise. The final two terms involve the resolutions of the toroid and beam position
monitors, indicated by σToroidResolution and σBPMResolution, respectively. The beam moni-
tors affect the lumi asymmetry resolution because they are used in the regression process,
discussed in the previous section. The degree to which the beam monitors contribute is
related by the experimentally determined correlation coefficients cn. The toroid term has a
slightly different form than that of the BPMs because it is used for both normalization and
as a regression parameter.
Each item in Equation 5.7 is separately known. The beam monitor resolutions can be
obtained directly from the data, as discussed in Section 3.2.2. Zero current beam pulses
occur at a rate of 0.5 Hz during normal running, allowing σPedestal to be measured reasonably
well. The contribution due to counting statistics is obtained from a Monte Carlo simulation.
The energy resolution of 50% is obtained from a simulation of the detector response. The
94
Luminosity Monitor Resolution ContributionsParameter Coefficient BPM Resolution Contribution (ppm)
Q -0.097 ppm/ppm 28.5 ppm 26E -3.17 ppm/MeV 2.0 MeV 6X -5.36 ppm/µm 3.3 µm 18Y -0.52 ppm/µm 4.5 µm 2dX -11.48 ppm/µR 0.1 µR 1dY 4.81 ppm/µR 0.2 µR 1
Counting Statistics 45Energy Resolution 20
Boiling Noise* 50Pedestal Noise 60
Total Calculated: 98Observed: 104
Table 5.1: Contributions to the luminosity monitor asymmetry resolution. *Target boilingis covered in Section 5.8.
energy resolution is low because many of the showers are not contained by the detector.
This leaves the term due to boiling noise, which is calculated using the method described in
Section 5.8. Combining these contributions for a run provides a full resolution accounting for
the lumi. Table 5.1 presents the resolution contribution itemization for a typical one-hour
data run.
The dominant contributions to the resolution are electronics noise, boiling noise, and
counting statistics. The remaining contributions due to beam monitor resolution are small
by comparison. The observed resolution of 104 ppm compares well with the calculated value
of 98 ppm, indicating that all major noise sources are known. The resolution exceeds the
200 ppm level required by the experiment.
5.8 Target Boiling
As discussed in Section 3.6, heating from the primary beam can cause density fluctuations
in the liquid hydrogen target. Though it is not technically “boiling” that is occurring, this
is still the standard name that has attached itself to this issue. The amount of boiling is
95
important to the experiment because it directly translates to noise for the detectors. Also,
if there exists an asymmetry in the density fluctuations, a false asymmetry measured by
the detectors could result.
In principle, once the lumi and Møller detector asymmetries are corrected for beam mo-
tion, the only correlation between them is in target density fluctuations and the resolutions
of the beam monitors used for the corrections. The latter contribution can be obtained
from the data, so by observing the level of common-mode noise between the two detectors,
the amount of boiling noise can be extracted. The method employed [42] begins by defining
two composite asymmetries through
J±≡AMøller±ALumi. (5.8)
Here A refers to the regression corrected asymmetries of the Møller detector and the lumi.
If the two detectors were completely uncorrelated, the RMS of the distributions of J+ and
J− would be the same. The amount that the two differ yields the common-mode noise
between the two detectors. Accounting for the noise due to beam monitor resolutions, the
amount of boiling noise is given by
σ2Boiling =
σ2J+
− σ2J−
4− (1 + cMQ + cLQ)σ2
ToroidResolution −5∑
n=1
cMn cLnσ
2BPMResolution. (5.9)
As with the preceding section, the cn terms refer to the beam correction coefficients. The
superscripts M and L refer to the Møller detector and the lumi, respectively. Similar
to Equation 5.7, the toroid term differs from the BPM terms because it is used for both
normalization and as a regression parameter.
To determine how much of this “boiling” noise is actually due to target density fluc-
96
Run Label Spot Size(mm x mm) Pump Speed(speed/nominal)1 1.5 x 1.5 12 1.0 x 1.0 13 1.0 x 1.0 14 1.0 x 1.0 15 1.5 x 1.5 2/36 1.0 x 1.0 2/37 1.0 x 1.0 2/38 1.0 x 1.0 2/39 1.5 x 1.5 1/310 1.0 x 1.0 1/3
Table 5.2: List of target boiling data runs.
tuations, as opposed to some other unmeasured common-mode noise source, a battery of
runs were taken in which the parameters of the beam and target were varied. Table 5.2
details the conditions for these runs, while Figure 5.14 displays the amount of boiling noise
calculated for each run, using Equation 5.9.
The term “spot size” refers to the dimensions of the beam at the entrance to the scatter-
ing chamber, as determined by the wire array (Section 3.4). Smaller beam spot size should
translate to greater target boiling effects. The vast majority of production data was taken
in the 1.0 mm x 1.0 mm spot size configuration. The pump speed regulates the velocity of
the liquid hydrogen around the target loop. Slower speeds should naturally translate into
larger boiling effects since the hydrogen remains in the beam for longer periods of time.
The nominal pump speed produces a liquid hydrogen velocity of 10 m/s, while the lowest
pump speed produces essentially zero flow [41].
Figure 5.14 shows little variation in the amount of calculated boiling noise among the
configurations tested. The small data spread indicates that the dominant common-mode
noise between the lumi and the Møller detector is not caused by target density fluctuations.
Even if all of the common-mode noise between the two detectors were due to boiling, the
effect is still only roughly 50 ppm. This amount of boiling noise is well below the proposal
97
Figure 5.14: Extracted boiling noise.
goal of 100 ppm [37]. Barring a large asymmetry observed by the lumi, this level of noise
renders boiling effects inconsequential for the Møller detector analysis.
5.9 Synchrotron Radiation Suppression
As discussed in Section 5.2, it is important that synchrotron radiation comprise no more
than 1% of the total lumi signal. Simulations show that the aluminum material in front of
the lumi reduces the synchrotron background by a large factor. To measure the suppression,
the signal sizes with the target in place and empty target runs are compared. When the
target is out, the dominant background is synchrotron radiation. Figure 5.15 displays the
synchrotron suppression results obtained for each chamber. The channel numbering follows
the convention established in Figure 5.5.
As expected from the synchrotron distribution (Figure 5.4) the background is heavily
peaked in chamber two, the channel on the right side of the lumi rings. Even if all of
this background were due to synchrotron photons, it is clear that it is significantly below
98
Figure 5.15: Synchrotron radiation background levels by chamber.
the desired 1% level. Because there are assuredly other background sources present, the
true synchrotron radiation background is actually closer to the difference between chamber
two and chamber zero, roughly 0.2%. The maximum asymmetry expected from synchrotron
radiation is at the 600 ppb level, so the asymmetry contribution of the synchrotron radiation
is reduced to the level of a few ppb.
5.10 Linearity
Although the lumi signal flux is quite large, the detector was designed to remain linear to
better than 1%. The main design concern is that electrons and ions could recombine before
being detected on the plates in the chambers. This effect would likely scale with signal size,
producing a non-linearity. The small distance between the collection plates was chosen to
limit electron transit time, and the 100 V bias was chosen to produce a fast signal. These
two design features limit the chance of electron-ion recombination, and are expected to
produce a very linear response.
99
5.10.1 Requirements
The requirement that individual systematic uncertainties be at the level of 5 ppb or below
can be used to set the linearity limit for the E158 detectors. For simplicity, the response of
the detector to charge is assumed to be a blend of linear and quadratic terms through
S = αF − βF 2, (5.10)
where S is the lumi signal size, F is the real signal flux, and α and β are constants. The
flux is related to the beam charge N through
FL,R = σL,RN, (5.11)
where L and R refer to the beam helicity, and σL,R is proportional to the scattering cross
section. The non-linearity of the detector response ε is defined as the ratio of the quadratic
term to the full signal, as
ε ≡ βF 2
αF − βF 2≈ βF
α. (5.12)
The E158 toroids are known to be very linear (Section 3.3.2), so their response T is assumed
to be simply proportional to charge:
T = N. (5.13)
Inserting the forms in Equations 5.10 through 5.13 into the asymmetry defined in Equa-
tion 5.2 yields
AMeasured = (1 − ε)APhys − εAToroid, (5.14)
100
where APhys refers to the true parity-violating asymmetry, and AToroid is the charge asym-
metry measured with the toroid. However, because charge is included as a regression pa-
rameter in Equation 5.3, the term proportional to AToroid is effectively removed. Note that
the charge regression coefficient can be identified as ε, the non-linearity of the detector. The
final result is then
AMeasured = (1 − ε)APhys. (5.15)
Assuming that the physics asymmetry is 15 ppb for the lumi and 150 ppb for the Møller
detector, the linearity 1− ε must be determined to 30% for the lumi and 3% for the Møller
detector, to keep the systematic uncertainty at the level of 5 ppb. Perhaps surprisingly, this
renders the linearity of the lumi unimportant.
The charge regression coefficient for the lumi varies between 3% and 8%. At face value,
this indicates that the lumi is less linear than expected. However, while the charge coefficient
certainly contains non-linearity information, as discussed in Section 6.7.2.2, it also contains
sensitivity to any unmeasured beam parameter. Therefore, while regression against charge
removes detector non-linearity, it can not be used to quantify it.
5.10.2 Measured Linearity
As demonstrated in the preceding section, it is not paramount that the lumi possess excep-
tional linearity, because of the additional regression performed against charge. However,
since the detector was designed to keep non-linearities below the level of 1%, it is still
instructive to quantify the linearity of the detector.
In principle, the linearity could be determined by observing the ratio of lumi signal to
the toroid signal for various currents, since the toroids are known to be essentially linear.
Figure 5.16 displays the lumi signal versus current for a single data run.
101
Figure 5.16: Lumi signal versus beam current.
Because most beam parameters such as position and angle are also correlated to charge,
Figure 5.16 is not a clean representation of the lumi response to current. The lumi is sensitive
to beam motion, so plots such as this can only be used to bound the lumi non-linearity to
∼10%, which is similar to the result obtained from the charge regression coefficient.
To obtain a better linearity determination, the effect of the motion of the beam must be
mitigated. One way to do this is to observe the ratio of the unnormalized lumi asymmetryA0
to toroid asymmetry AToroid. The asymmetries are defined as
ALumi0 ≡ SR − SL
SR + SL(5.16)
and
AToroid ≡ TR − TL
TR + TL, (5.17)
with S and T referring to the lumi and toroid signals, respectively. Using the forms for the
lumi and toroid signals given in Equations 5.10 and 5.13, the relation between ALumi0 and
102
AToroid is
ALumi0 = (1 − ε)AToroid. (5.18)
The ε term that appears on the right-hand side of Equation 5.18 modifying the proportion-
ality to the toroid asymmetry is again the lumi non-linearity.
The linearity is determined by plotting the ratio of ALumi0 to AToroid as a function of the
analysis cut on beam jitter. As the beam cuts are tightened, the ratio of the uncorrected
lumi asymmetry to the toroid asymmetry then tends toward 1 − ε, the linearity of the
detector.
Figures 5.17 and 5.18 show the results of measuring this ratio as a function of the cut on
beam jitter. Here 100 production data runs have been used, which is roughly one quarter
of the full data set for Run I. The beam cut refers to the maximum allowed beam jitter,
in both X and Y, measured at the lumi. These numbers were obtained using the lumi as a
BPM, as detailed in Section 5.12. The points at the left of the plot have the most stringent
beam jitter cut.
Figure 5.17: Full range of results.
It is clear that the ratio never departs dramatically from unity, indicating that no gross
103
Figure 5.18: Tight beam cuts subset of linearity results.
non-linearity exists. As the beam cuts are tightened, the ratio is seen to vary between 98.0%
and 100.5%, demonstrating the linearity of the lumi to the level of ≈ 2%.
5.11 Missing Pulse Effect
An unexpected feature of the lumi signal reveals itself after a beam pulse is absent. Missing
pulses occur regularly at 0.5 Hz, due to a scheduled pedestal pulse, and random beam drop-
outs also occur. Figure 5.19 displays the charge-normalized front lumi signal versus time,
with a missed-pulse occurring at the zero of the x-axis. The signal is noticeably larger after
the missed pulse, though it is still only 0.5% above the average signal. The effect decays
quickly and is virtually absent after 3 pulses.
A possible explanation for this behavior is that the detector, which is really a capacitor, is
only allowed to recharge fully during the missing pulse. This would explain why subsequent
pulses receive a boost in size. No other detectors see the effect, so it appears that this
behavior is internal to the lumi design.
Whatever the true cause, the effect produces additional tails to the lumi asymmetry
104
Figure 5.19: Charge normalized lumi signal following a missing pulse.
distributions, resulting in an effective decrease in the asymmetry resolution. To combat
this, a cut in the analysis is made to remove the first 8 pulses after the pedestal pulse.
This amounts to a 7% cut on the data. Figure 5.20 demonstrates the improved detector
resolution when the additional cut is included.
Figure 5.20: The lumi asymmetry distribution without (left) and with (right) a cut after amissing pulse. RMS is the statistical width of the distribution, while Sigma refers to thewidth determined from the fit.
This cut renders the data sets of the lumi and Møller detector slightly different. So that
the lumi can still serve as a viable systematic cross-check, it was verified that the cut does
105
not change the mean returned by the Møller detector at a significant level.
5.12 Lumi as a BPM
There are no amplifiers used for the lumi chambers, and all chambers are identical, making
the relative gain among the chambers very close to one. An interesting application of this
feature allows the lumi to be used as a beam position monitor.
The X and Y positions at the lumi are calculated by weighting the chamber signals S
by the chamber spatial positions through
X = αx
∑8n=1 sin θSn∑8n=1 Sn
(5.19)
and
Y = αy
∑8n=1 cos θSn∑8n=1 Sn
, (5.20)
where θ refers to the azimuthal location of the chamber. By convention, zero degrees refers
to the chamber at the top of the lumi ring, and the angle increases in the clockwise direction
looking downstream from the target.
The α coefficients are determined by using the beam position in the BPMs upstream of
the target and extrapolating to the position at the lumi detector. Several runs were taken
with large angular displacement on target during the course of the experiment, allowing for
a large lever-arm to determine the α coefficients. Figure 5.21 displays the results obtained
utilizing these runs. Each point represents the average position over the course of one full
run.
The displacements used in these plots are an order of magnitude greater than would be
106
Figure 5.21: Calculated beam position using the lumi, versus position position at lumifigured using angle and position bpms.
expected naturally during a typical data run. It is clear that the lumi response to smaller
displacements should be linear. Note that the α coefficients are different for X and Y
because the beam was not centered through the detector.
The main utility of using the lumi as a beam monitor is that it provides the beam
location at the position of the detectors. Having an effective BPM was particularly useful
when adjusting the dipole chicane magnets, which are downstream of the last E158 BPMs.
Also, it was simple to create a display that continually updated the position reported by
the lumi, so that data takers could quickly see that the beam was well positioned through
the detectors. Moreover, since the detector gains were found to be very stable, the lumi
proved to be a useful tool for beam re-alignment after long beam-off periods.
107
Chapter 6
Asymmetry Analysis
This chapter describes the determination of the parity-violating asymmetry APV from the
raw asymmetry ARaw measured with the Møller detector. The effects of background cor-
rections and dilutions are discussed, and the systematic uncertainty on the measurement is
estimated. The analysis of the luminosity monitor data is also presented.
6.1 Initial Processing
Before any analysis is performed, an initial pass is made over the data to put it into a usable
form. The 0.5 Hz no-beam pulses are used to subtract the pedestal from all detector and
beam monitor channels. The subtraction for a given beam pulse utilizes the running average
of the preceding 10 pedestal pulses. The data from the Møller detector is also “blinded”
with a random offset, such that the asymmetry A from each channel becomes
A→ A+A′, (6.1)
where A′ is a single random number in the range of ±200 ppb. The value of A′ was not
revealed until the asymmetry analysis was completed. The offset was included to reduce
the potential for human bias in the determination of APV .
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6.2 Detector Channel Weights
The raw pulse-pair asymmetry ARaw measured with the Møller detector is defined as a
weighted average of the detector channels, given by
ARawi =30∑j=1
wjAji . (6.2)
Here Aji denotes the asymmetry measured with the jth detector channel for the ith pulse
pair, and wj is the weight for that channel. The weights wj are constrained by
30∑j=1
wj = 1. (6.3)
The sum covers the 30 channels of the Møller detector, comprised of the In and Mid regions
(Section 4.1) of the E158 calorimeter depicted in Figure 4.4.
The weights are found on a per-run basis through a minimization method which takes
into account inter-channel correlations. The method is employed to maximize the asym-
metry resolution of the overall Møller detector. First, the symmetric matrix Mij is defined
by
Mij = 〈AiAj〉 − 〈Ai〉〈Aj〉, (6.4)
where An denotes the asymmetry measured with channel n. Channel weights are defined
by minimizing the sum N , defined as
N =30∑i=1
30∑j=1
wiwjMij . (6.5)
Note that in the absence of correlations among the channels in Equation 6.4, the weights
109
reduce to
wi =1σ2
i∑30j=1
1σ2
j
, (6.6)
the standard statistical weights.
6.3 Regression
The sensitivity of the detectors to beam motion and energy jitter is reduced through a
process called regression. Figure 6.1 depicts the raw asymmetry of the Møller detector
(Equation 6.2) versus the Y position asymmetry, measured with a BPM just upstream of
the target. The strong correlation indicates that asymmetries in the Y position can be
manifested in the asymmetry measured with the Møller detector.
Figure 6.1: Møller detector asymmetry versus Y position asymmetry.
The regression procedure involves calculating the correlation slope seen in Figure 6.1
and removing it on a per-pulse basis. The detector asymmetry ARaw is corrected for the
beam asymmetry ∆Bn through
A = ARaw −6∑i=1
cn∆Bn, (6.7)
110
where A is the corrected asymmetry, and the sum runs over the beam parameters of charge
Q, energy E, X and Y position, and X and Y angle. The regression coefficients cn are the
correlations of the detector asymmetry with the nth beam parameter.
The coefficients are experimentally determined by performing a multidimensional fit of
ARaw against the six beam parameters. The covariance matrix Bij is defined as
Bij = 〈∆Bi∆Bj〉 − 〈∆Bi〉〈∆Bj〉, (6.8)
and the vector Vi is
Vi = 〈Araw∆Bi〉 − 〈ARaw〉〈∆Bi〉. (6.9)
The coefficients cn are obtained by inverting the matrix B and multiplying by the vector
V .
cj =6∑i=1
B−1ij Vi (6.10)
Because the coefficients could change over time, they are determined every 10,000 pulse
pairs.
Figure 6.2 depicts the same data presented in Figure 6.1 after the regression correction
of Equation 6.7 has been applied. The correlation of the detector asymmetry with the beam
asymmetry is greatly reduced.
Regression also enhances detector resolution by reducing the effect of beam fluctuations
on the asymmetry distribution measured with the detector. Figure 6.3 depicts the im-
provement in detector performance with regression. The resolution is improved by ≈ 60%.
The different E158 detectors are positioned at different geometries, causing their sensi-
111
Figure 6.2: Møller detector asymmetry versus Y position asymmetry after regression.
Figure 6.3: Møller detector resolution with and without regression.
tivity to beam parameters to vary greatly. Table 6.1 presents the coefficients of three E158
detectors, averaged over all Run I data. The Out detector is the most sensitive detector
for all categories. The large coefficients are due to the large slope in its signal flux profile
(Figure 3.25), so that small deviations in beam parameters result in comparatively large
changes in the Out detector signal size.
112
Run 1 DetectorCoefficient Møller Out Front Lumi
Q (ppm/ppm) 0.005 -0.08 -0.03E (ppm/MeV) -25.9 76.2 -5.1X (ppm/µm) 0.5 6.8 -1.1Y (ppm/µm) -1.2 2.0 0.4dX (ppm/µR) -66.1 128.2 25.2dY (ppm/µR) 7.3 65.5 -12.9
Table 6.1: Regression coefficients for Run I.
6.4 Beam Dithering
Beam dithering is an alternative method to regression for correcting detector asymmetries.
The coefficients of Equation 6.7 are found by intentionally moving the beam with corrector
magnets and observing the detector correlation with the beam monitors. Figure 6.4 depicts
Figure 6.4: Location of components used for beam dithering.
the location of the linac components used for beam dithering. Corrector magnets located in
the final sectors of the linac are used to vary both beam position and angle, by an amount
several times the natural beam jitter. The energy is varied by adjusting the phase of a
klystron in the same region. Figure 6.5 depicts the response of one of the lumi chambers to
an x-position dithering cycle1.
The beam parameters were typically dithered 4% of the time. This data is only used
for the determination of correction slopes and is removed from the production data. The1The slowly varying dither cycle depicted in the figure was found to be very disruptive to the beam
feedbacks. For the production data, a more rapid dithering style was employed.
113
Figure 6.5: A lumi chamber responding to a position dither cycle.
response of the detectors to the dithered beam parameters is found in a manner similar to
the regression method, by a multidimensional fit to the data in which dithering is occurring
(Equation 6.10). The detector asymmetry is then corrected as in Equation 6.7. Because
both the lumi and Out detector require a correction for charge, a regression against charge
is done after the other beam coefficients are computed.
In contrast to the regression procedure, dithering varies the beam parameters in a charge
independent way. By decoupling the parameters, the dithering results are less prone to
systematic effects. However, dithering was not always functional, while the regression pro-
cedure can always be performed. Therefore, the main results for the E158 experiment are
computed using regression, while the dithering results are used as a cross-check.
6.5 Data Selection
The quality of the data used in the asymmetry analysis is ensured by removing pulses that
do not pass predetermined criteria [39]. To avoid biasing the data, many of the data cuts
also remove a certain number of pulses before and after an offending pulse. The following
paragraphs describe each of the requirements employed in the asymmetry analysis.
114
General: The primary quality cut is composed of several separate cuts that are always
made in the analysis. The first ensures that the beam parameters of charge, energy, X
and Y position, and X and Y angle are within six standard deviations of the mean value.
The second removes all data with large excursions due to beam dithering. The third cut
ensures that the time between beam pulses is the same for each pulse in the pulse pair, to
combat possible hysteresis effects. The final cut requires that the agreement of the charge
asymmetry measurements of the toroids upstream of the target be within 1000 ppm, roughly
20 standard deviations. Any pulse pair not meeting these criteria is removed, as well as the
50 preceding and subsequent pulse pairs. Overall, this cut retains 90.8% of the data.
Position Jitter: Pulse pairs with position jitter greater than nine standard deviations
from zero are removed, as well as the 50 preceding and following pulse pairs. The cut has
a 91.9% retention rate.
Energy Stability: The momentary loss of a klystron in the linac is not uncommon. De-
pending on the phase of the missing klystron, the energy of the beam can either increase or
decrease, usually by an amount of ≈ 200 MeV. Off-energy pulses are removed by checking
for large deviations in the E158 energy BPM. The cut has a window of ± 50 pulse-pairs,
and has a retention rate of 97.7%.
BPM Phase: The resolution of the BPMs decreases when the phase of the local oscillator
drifts. This cut uses the Q cavity of each BPM to ensure that the phase is within an
acceptable range. The time structure for the phase drifts causes this cut have a window
from 5 pulse pairs before the offending pulse to 50 pairs after. The retention rate is 97.4%.
115
BPM Linearity: Large offsets, on the scale of a few millimeters, cause the BPM response
to become non-linear. This cut ensures that the beam position is within a range where the
BPMs are linear to better than 99%. The cut window is from two pulse pairs before the
offending pulse to 4 pulse pairs after. The retention rate is 97.0%.
Charge: The beam charge is required to be greater than 1011 electrons per pulse, about
20% of the normal beam current. Pulses with low charge are fundamentally different from
the standard beam pulses, and are excluded. The cut only removes one pulse pair. The
retention rate is 98.1%.
Transmission: The transmission from the source to the E158 target is required to be
greater than 90%. Like low-current pulses, low-transmission pulses are very different from
normal beam pulses and are removed. The retention rate is 99.9%.
Source Voltages: Data in which the CP and PS Pockels cell voltages are improperly set
at the source are excluded by this cut. When the voltages are incorrect, the polarization of
the beam is unknown. The retention rate is 99.3%.
Timeslot: While the beam repetition rate is 120 Hz, the electric power for the accelerator
is at the normal 60 Hz rate. The position of the beam pulses in time relative to the phase
of the electric power can therefore assume two states, which are called timeslots. Because it
is possible that the two timeslots have different properties, this cut requires that each pulse
in a pair occur in the same timeslot. The retention rate is 98.9%.
Slopes: The regression coefficients used to correct the detector asymmetries are calculated
in chunks for 10,000 pulse pairs, sorted by timeslot. It is possible that one timeslot can
dominate a period of time so that the other timeslot has very few pulses. If the smaller
116
timeslot has less than 100 pairs, this cut removes that timeslot because its regression slopes
would be unreliable. The retention rate is 99.2%.
All Cuts The total data retention rate when all cuts are employed is 73.5%.
Careful studies were done on the Run I Møller detector asymmetry result in which cuts
were included or removed to observe their effect on the mean asymmetry. In all cases it was
found that the cuts do not bias the data in any discernible way.
6.6 Møller Detector Asymmetry Analysis
The regression corrected Møller detector asymmetry distribution defined by Equation 6.7
is used to determine a mean asymmetry Ai and statistical uncertainty σi for each data run
consisting of roughly 200,000 pulse pairs. The results of N individual data runs can be
combined to compute the average asymmetry Aaverage and uncertainty σaverage, with
Aaverage =
∑Ni=1
Ai
σ2i∑N
i=11σ2
i
(6.11)
and
σaverage =
√√√√ 1∑Ni=1
1σ2
i
. (6.12)
Figures 6.6 and 6.7 present the results obtained with the Møller detector for all of Run
I and Run II, respectively. The data has been averaged into periods of data called “slugs,”
which are runs with the same beam energy and source halfwave plate setting. (The different
energy and halfwave plate states are used to combat systematic effects, and are described
in the following section.) Both Run I and Run II are comprised of approximately the same
amount of data, though Run II has less systematic reversals.
117
Figure 6.6: Run I Møller detector asymmetry versus slug.
The Run I and Run II plots have reasonable χ2/ndf and their means agree within
statistical fluctuations. The clear asymmetry in both plots represents the first observation
of parity violation in Møller scattering.
The data plotted in Figures 6.6 and 6.7 are purely statistical averages that do not include
systematic uncertainties such as beam polarization, regression correction uncertainties, and
background corrections. These issues are discussed in Section 6.8 to arrive at APV .
6.6.1 Systematic Reversals
The experiment utilized two distinct methods to reduce sensitivity to helicity-correlated
systematic effects [37]. The first involved running with two different source halfwave plate
states (Section 3.1.4). Inserting the waveplate reverses the helicity of the laser light hitting
the cathode while keeping the rest of the source setup the same. The second method
involved running at two separate beam energies, 45 GeV and 48.3 GeV. The difference in
these energies represents a 180 g-2 rotation of the electron spin as it traverses the A-Line
118
Figure 6.7: Run II Møller detector asymmetry versus slug.
bend leading up to End Station A.
The four possible apparatus configurations combine the physics asymmetry APV with
two types of helicity-correlated systematic effects denoted as A1sys and A2
sys. The asymmetry
A1 refers to systematic effects that are not affected by the waveplate state, such as electronics
cross-talk with the bias voltage of the CP cell at the source (Section 3.1). The asymmetry
A2 refers to effects like residual linear polarization in the laser light at the source that can
also reverse sign with the insertion of the source halfwave plate. The four configurations
are then given by
A45Out = APV +A1
Sys +A2Sys
A45In = APV −A1
Sys +A2Sys
A48Out = APV −A1
Sys −A2Sys (6.13)
A48In = APV +A1
Sys −A2Sys,
119
(6.14)
where AEnergyHalfwave denotes the overall asymmetry seen with the Møller detector for each
energy and halfwave plate setting. By collecting the same amount of data in each state,
systematic effects are minimized. Run II had less systematic reversals than Run I and will
be shown to have larger systematic effects, discussed in Section 6.7.2.2.
Figure 6.8 presents the Møller detector asymmetry obtained for each energy-halfwave
plate configuration in Run I and Run II. All of the states agree well, indicating that there
are no large systematic effects. Figures 6.9 and 6.10 present the same data as the previous
plots, but without the systematic reversal sign flips taken into account. They provide
another visual indication that systematic uncertainties are below the level of statistical
fluctuations for the Møller detector.
Figure 6.8: Measured asymmetry for each energy-halfwave plate setting.
6.6.2 Beam Corrections
Because beam monitors are used to modify the raw Møller detector asymmetry through
Equation 6.7, beam asymmetries lead to a correction of the detector asymmetry. Table 6.2
120
Figure 6.9: Run I Møller detector asymmetry versus slug, sign flips suppressed.
Run IParameter Correction (ppb)
Charge 0.2Energy 5.1
X 6.7Y 0.05
X Angle 23.8Y Angle 2.0Total 37.9
Run IIParameter Correction (ppb)
Charge -1.8Energy 29.0
X -3.7Y -10.5
X Angle 17.1Y Angle -8.9Total 21.2
Table 6.2: Beam corrections to ARaw of the Møller detector.
lists the beam corrections for Run I and Run II. Note that the corrections have no un-
certainty: they are exactly the amount the beam monitors have shifted the raw Møller
detector asymmetry. The contribution to the uncertainty on the Møller asymmetry enters
through the systematic uncertainty associated with the method itself and is discussed in
Section 6.7.2.2.
The agreement between the corrected detector asymmetry obtained with regression and
dithering is an important measure of potential systematic uncertainties. Table 6.3 com-
pares the asymmetries obtained with the two methods. The values differ slightly from the
121
Figure 6.10: Run II Møller detector asymmetry versus slug, sign flips suppressed.
Run I Asymmetry (ppb) Correction (ppb)Regression -179.1 ± 24.3 30.6Dithering -182.2 ± 27.0 27.5Run II Asymmetry (ppb) Correction (ppb)
Regression -150.0 ± 22.3 11.5Dithering -154.8 ± 22.7 6.7
Table 6.3: Comparison of regression and dithering results for the Møller detector asymmetry.
averages shown in Figure 6.6 and 6.7 because only the subset of data in which dithering
was functional was used. The improvement in the Møller detector resolution is less for
dithering, as indicated by the uncertainty on the asymmetry. The most likely cause is that
the dithering correlation slopes are computed less often than the regression slopes. In both
Run I and Run II, the methods agree to the level of a few ppb, indicating that systematic
differences between the methods are small. However, Section 6.7.2.2 will demonstrate that
the total systematic uncertainty contribution to APV due to beam corrections is dominated
by a systematic effect common to both regression and dithering.
122
6.7 Asymmetry Corrections and Uncertainties
The parity-violating asymmetry APV is obtained from the beam corrected Møller detector
asymmetry AMeasured by accounting for contributions from backgrounds and taking into
account scale factors. The asymmetries are related through
APV =1Pλ
AMeasured −∑i fiAi1 −∑i fi
, (6.15)
where the sums are taken over all backgrounds. The fi terms represent the dilution factors of
the backgrounds, defined as the ratio of the background signal to the total signal seen in the
Møller detector. The helicity-dependent asymmetries of the backgrounds are represented
by the Ai terms. The scale factors are the beam polarization P and the linearity of the
detector λ. Sections 6.7.1 through 6.7.8 detail the components of Equation 6.15 for the
Møller detector. Section 6.8 then uses the results to compute APV for Run I and Run II.
6.7.1 The Electron-Proton Scattering Correction
The spectrometer (Section 3.7) is designed to provide separation of electron-electron and
electron-proton (eP) scattering events. However, the separation is not perfect, and some
electron-proton events are registered in the Møller detector. In particular, inelastic eP
events are expected to have an asymmetry an order of magnitude larger than APV . It is
therefore crucial to account for the eP background contamination in AMeasured.
The amount of eP signal in the Møller detector is determined from a Monte Carlo sim-
ulation of the E158 spectrometer [43]. The simulation is configured to match the maps of
the signal flux versus radius, provided by the profile monitor (Section 4.3). Figure 6.11
depicts a comparison between the simulation and a profile scan with typical running condi-
123
tions. Figure 6.12 presents a scan with the insertable acceptance collimator (Section 3.7.4)
in place. The additional separation of the Møller from the electron-proton scattering events
is important for ensuring that the simulation matches the reality of the signal distribution.
In both plots, the agreement between the simulation and the actual data is reasonable.
Figure 6.11: Data and simulation comparison with normal running conditions.
Because the difference between the data and the simulation is small, the simulation can
be used to quantify the eP contamination in the Møller detector. Figure 6.13 displays a
profile scan with the electron-electron and electron-proton contributions separated in the
simulation.
Table 6.4 presents the dilution factors f used for determining the correction to AMeasured
for the electron-proton contribution in Equation 6.15. The ratio of elastic to inelastic events
R is also shown. The uncertainty on the results was determined by varying the input
parameters of the simulation within a reasonable range.
The Run I and Run II results differ for two reasons. First, the eP collimator (Sec-
124
Figure 6.12: Data and simulation comparison with the insertable acceptance collimator inposition.
tion 3.7.5) was only used in Run II. The collimator blocks the eP detector and most of
the Out ring. Second, the quadrupole magnet settings of the spectrometer were changed
between Run I and Run II to de-emphasize the Out ring of the detector.
The value of AMeasured must also be corrected for the asymmetry contribution of the
electron-proton scattering background. The asymmetry result of the eP detector in Run I
is used as an input to the simulation. The flux in the eP detector is approximately 70%
Run I 45 GeV 48 GeVDetector f σf R f σf R
In 0.0969 0.0099 7.4 0.0861 0.0087 6.8Mid 0.0684 0.0079 5.7 0.0610 0.0071 5.1Out 0.1401 0.0203 4.4 0.1903 0.0261 4.0
Run II 45 GeV 48 GeVDetector f σf R f σf R
In 0.0810 0.0095 7.1 0.0780 0.0079 6.1Mid 0.0540 0.0067 5.8 0.0520 0.0053 5.5Out 0.1020 0.0143 4.9 0.0810 0.0153 4.4
Table 6.4: Dilution factors f due to background eP scatters. R is the ratio of elastic toinelastic eP signals.
125
Figure 6.13: Profile scan with the Møller and electron-proton scattering simulation resultssuperimposed.
Run I 45 GeV 48 GeVDetector fA σfA fA σfA
In -33.4 4.7 -35.5 4.8Mid -32.2 5.2 -33.8 5.8Out -86.9 22.3 -140.2 33.1
Run II 45 GeV 48 GeVDetector fA σfA fA σfA
In -29.4 4.6 -33.2 4.6Mid -26.5 3.8 -27.5 3.8Out -57.5 9.0 -52.4 9.2
Table 6.5: Diluted asymmetries fA due to the electron-proton scattering background. Allentries are in ppb.
elastic eP scatters, 27% inelastic eP scatters, and only about 3% Møller scattered electrons.
However, the asymmetry associated with the inelastic eP events is roughly 25 times larger
than the elastic asymmetry, so these events dominate.
Figure 6.14 depicts the asymmetry measured with the eP detector for Run I. The Q2 is
sufficiently different between the two beam energies that they are averaged separately. The
result is consistent with the theoretical estimate [68].
Table 6.5 details the diluted asymmetry fA due to the electron-proton scattering back-
ground used to correct AMeasured. The large uncertainty on the Run I result for the Out
ring is dominated by the simulation of the shower sharing with the eP detector. In Run II,
the eP collimator greatly suppresses the effect, reducing the systematic uncertainty.
126
Figure 6.14: Asymmetry result from the eP detector in Run I.
6.7.2 Beam Asymmetry Correction Systematic Uncertainties
The regression corrections given in Equation 6.7 must be assigned a systematic uncertainty
contribution to APV . Examining Equation 6.15, the uncertainty can be accommodated
formally as a correction with zero dilution f or asymmetry fA, but with an uncertainty
σfA.
The systematic uncertainty estimate is divided into two pieces: first-order and higher-
order effects. The following sections will demonstrate that while the In and Mid regions of
the detector are fairly insensitive to the beam correction systematic effects, the Out ring
is dominated by the higher-order effects. Because of the large uncertainty from this effect,
the Out ring is not included with the In and Mid ring in the calculation of APV .
6.7.2.1 First-Order Beam Correction Systematic Uncertainties
The first-order systematic uncertainty assigned to the values in Table 6.2 is found by en-
hancing the Møller detector sensitivity to the regression parameters by re-weighting the
127
Regression CoefficientsParameter Monopole X-Dipole Y-Dipole
Q (ppm/ppm) 0.005 0.012 0.005E (ppm/MeV) -25.9 -2.6 -10.0X (ppm/µm) 0.5 -22.1 -0.8Y (ppm/µm) -1.2 -1.6 24.0dX (ppm/µR) -66.1 -30.8 -31.0dY (ppm/µR) 7.3 18.9 151.7
Table 6.6: Regression coefficients of three Mid detector weighting schemes.
detector channels. In general, the asymmetry A for any weighting pattern is defined as
A =1√
N∑Ni=1w
2i
N∑1
wnAn, (6.16)
where An refers to the asymmetry of channel n, with weight wn [70]. The sums are taken
over the N channels of the detector. The most relevant weighting scheme is the dipole
pattern, with weights given by
wDipolen = sin(2πnN
). (6.17)
Assigning n = 0 as the top of the detector, this scheme produces enhanced sensitivity
to beam motion in the horizontal direction and is known as the X-Dipole pattern. Rotating
by 90, the pattern becomes a Y-Dipole. Table 6.6 displays the regression coefficients of
the Mid ring Monopole, Mid X-Dipole, and Mid Y-Dipole weighting schemes.
The monopole weighting scheme is dominated by the energy parameter, while the X
and Y dipoles are dominated by the spatial parameters their names suggest. Owing to their
increased sensitivity, specific weighting schemes are employed to estimate the systematic
uncertainty inherent in the regression method. Table 6.7 relates the patterns that were
especially sensitive to each of the beam parameters2.2The Q parameter is excluded in the first-order systematic uncertainty estimate. It is more relevant for
the higher-order uncertainty discussed in Section 6.7.2.2.
128
Sensitive Weighting SchemesParameter Detector
E Møller MonopoleX Mid X-DipoleY Mid Y-DipoledX Out X-DipoledY Out Y-Dipole
Table 6.7: Sensitive pattern-weighted detectors for the regression beam parameters.
The first order systematic uncertainty on the beam corrections is estimated by using
the sensitive monitors in Table 6.7 to compute the relative error εn on each of the n beam
corrections. Two separate sets of data, denoted by the superscripts, are used to determine
the relative uncertainty through
εn =A1n −A2
n
corr1n − corr2n→ ∂An
∂corrn, (6.18)
where An is the mean asymmetry of the detector weighting pattern sensitive to the nth
beam parameter, and corrn refers to the beam correction of An. To ensure that first order
effects dominate, the two data sets are chosen as the two timeslots in the data slug with
the largest difference in the beam corrections.
The first-order systematic uncertainty δn for the correction corrn is the product of the
relative uncertainty with the value of the correction:
δn = εncorrn. (6.19)
Tables 6.8 and 6.8 display the systematic uncertainty calculated for each regression
parameter using this method for both Run I and Run II. To be conservative, the value
quoted as the relative error is the magnitude of the error found using Equation 6.18, plus
one standard deviation. The uncertainty on the total correction is found as the quadrature
129
Run IParameter Correction (ppb) Relative Error ε (%) Uncertainty δ (ppb)
E 5.1 7.0 0.36X 6.7 8.7 0.58Y 0.1 10.1 0.01dX 23.8 3.9 0.92dY 2.0 22.3 0.45
Total: 1.2
Table 6.8: Run I first-order systematic uncertainties in the regression corrections to the rawMøller detector asymmetry.
Run IIParameter Correction (ppb) Relative Error ε (%) Uncertainty δ (ppb)
E 29.0 10.0 2.00X -3.7 3.8 0.17Y -10.5 2.8 0.31dX 17.1 3.6 0.69dY -8.9 5.7 0.55
Total: 3.0
Table 6.9: Run II first-order systematic uncertainties in the regression corrections to theraw Møller detector asymmetry.
sum of the individual uncertainties. In both data sets, the total first-order systematic
uncertainty is small.
6.7.2.2 Higher-Order Beam Correction Systematic Uncertainties
The plots of the Møller asymmetry versus slug shown in Figures 6.6 and 6.7 have χ2/ndf
near unity, and the systematic reversal plots of Figure 6.8 show no indication that large
systematic uncertainties are present. However, when the Out ring is examined in the same
manner, it is clear that there are possible systematic effects that need to be addressed.
Figures 6.15 and 6.16 present the Out ring asymmetry data. The Run I slug plot
has an elevated χ2/ndf, but still looks reasonable. The Run I systematic reversal plot
appears to have some cancellation of an effect between the 48 GeV-Halfwave Out and 45
GeV-Halfwave Out detector results. In the Run II plots, however, it is clear that there
130
are systematic effects not removed by the regression method. The degree to which this
unregressed systematic effect can influence the Møller detector asymmetry must be assigned
a systematic uncertainty.
Figure 6.15: Run I Out ring asymmetry data.
Figure 6.16: Run II Out ring asymmetry data.
The nature of the systematic effect was isolated in Run III of the E158 experiment. In
addition to the normal setup, the BPM signals were also “sliced” in time and fed to four
different ADC channels. That is, the first quarter of the pulse went to the first channel,
and so on. Effectively, this allows for regression against the shape of the beam pulse. The
Run III Out ring data look similar to Run II, with poor fits when the normal analysis is
employed. However, when regression against the sliced BPM signals is included, the Out
131
data are greatly improved, with a χ2/ndf near unity [72]. Therefore, the cause of the Out
detector systematic effect is found to be intra-pulse helicity-correlated asymmetries which
are not removed by the normal regression procedure in Run I and Run II.
The nature of the effect can be understood by the following simple description. The
BPMs measure the average position X of the electron beam pulse3, given by
X =1T
∫ T
0x(t)dt, (6.20)
where the integral is over the duration of the pulse T and x is the beam position. The
regression procedure then corrects the raw detector asymmetry ARaw by multiplying the
beam difference ∆X by a constant coefficient c, as in Equation 6.7. The correction C is
then given as
C =c
T
∫ T
0x1(t) − x2(t)dt, (6.21)
where the subscripts refer to the beam pulses of the pulse pair. The method is valid unless
the detector sensitivity to the beam varies over the course of the beam pulse. In that case,
the constant c becomes a function of position, and actually needs to be integrated as
C =1T
∫ T
0c(x1)x1(t) − c(x2)x2(t)dt. (6.22)
Effectively, this is what “slicing” the BPM signals does: it allows for the detector sensitivity
to vary over the course of the beam pulse.
A possible mechanism for this to occur is known as “tail wagging,” where the tail of
the beam pulse has large fluctuations due to the passage of the head of the pulse through3Technically, the BPMs measure the charge-averaged position. The formula could be corrected by x(t)
→ ρ(t)x(t), where ρ is the charge density. However, for clarity, the charge dependence is omitted.
132
the linac. This is known as the “wakefield effect,” where the electromagnetic “wake” of the
head of the pulse disrupts the tail [73, 74]. The effect could be on the scale of hundreds
of microns, which would indeed produce large effects in the Out ring, due to its sensitive
geometry [75]. If the intra-pulse effects are also helicity-correlated, the asymmetry results
are not fully corrected, as seen in the Out ring results in Figure 6.15 and 6.16.
Because there were no “sliced” BPM signals in Run I and Run II, the systematic un-
certainty due to higher-order effects can only be estimated. Because the plots of the Møller
detector asymmetry versus slug look normal, it is expected that the sensitivity to higher-
order effects should be small. First, it is assumed that the measured asymmetry AMeasured
is related to the real asymmetry AReal through
AMeasured = AReal + αAsys, (6.23)
where Asys is the asymmetry due to beam-related systematic effects, and α is the detector
sensitivity to Asys [76]. If two detectors are examined, Equation 6.23 can be used to find
α1Asys ≡ σ1
sys =α1
α1 − α2((AMeasured
1 −AMeasured2 ) − (AReal1 −AReal2 )), (6.24)
where the subscripts denote the detectors and σ1sys is the systematic uncertainty assigned
to detector 1. If we choose the Møller detector as the first detector and the Out ring as the
second, we have
AReal1 −AReal2 = ∆eP, (6.25)
where ∆eP represents the difference in the correction for the electron-proton scattering
background between the two detectors. The only unknown values are then α1 and α2,
133
Systematic Uncertainty Estimates (ppb)Run I Run II
∆eP 60.6 ± 26.1 21.1 ± 5.6AMeasured
1 −AMeasured2 -69.7 ± 46.5 -104.1 ± 48.5
α1α2
-0.064 -0.058σMøllersys -7.8±3.2 -6.9±2.7
Table 6.10: The higher-order systematic uncertainty computed for the Møller detector,comparing the Møller detector (1) and the Out ring (2), assuming Equation 6.26.
which can only be estimated.
One method is to note that the Out ring has a large charge regression coefficient (Ta-
ble 6.1), between five and ten times larger than would be expected (Section 5.10.1). Because
most beam parameters are correlated to charge, it is reasonable to assume that the charge
coefficient is a measure of the size of the systematic effect.
As an estimate, it is assumed that the α systematic coefficient is proportional to the
charge regression coefficient cQ. The ratio of the sensitivity to the systematic effect of the
two detectors is then
α1
α2=c1Qc2Q. (6.26)
With Equation 6.25 and the assumption of Equation 6.26, the systematic uncertainty con-
tribution for the Møller detector can be found with Equation 6.24. Table 6.10 represents
the Møller systematic uncertainty found for Run I and Run II using this method.
Because Equation 6.26 is an assumption, it is useful to estimate the ratio α1α2
by an
alternative method. The uncertainty on AMeasured in Equation 6.23 is
σ2AMeasured = σ2
AReal + α2σ2Asys. (6.27)
134
Systematic Uncertainty Estimates (ppb)Run I Run II
∆eP 60.6 ± 26.1 21.1 ± 5.6AMeasured
1 −AMeasured2 -69.7 ± 46.5 -104.1 ± 48.5
α1α2
0 ± 0.062 -0.141 ± 0.016σMøllersys 0 ± 8.1 -15.5 ± 6.2
Table 6.11: The higher-order systematic uncertainty computed for the Møller detector,comparing the Møller detector (1) and the Out ring (2), using Equation 6.29.
Again using two detectors denoted by 1 and 2, the ratio of the α coefficients is
∣∣∣∣α1
α2
∣∣∣∣ =√√√√√σ2
AMeasured1
− σ2AReal
1
σ2AMeasured
2− σ2
AReal2
, (6.28)
which can be approximated as
∣∣∣∣α1
α2
∣∣∣∣ = σAMeasured1
σAMeasured2
√√√√χ2ndf1 − 1χ2ndf2 − 1
. (6.29)
Geometry causes the regular regression coefficients for the Møller and Out detectors
to be anti-correlated, so it is reasonable to assume that α1α2
is negative. When one of the
detectors has a χ2ndf < 1, the method breaks down, because no systematic effects can be
discerned. Therefore, it is also useful to calculate the uncertainty on Equation 6.29:
σ
(α1
α2
)=σAMeasured
1
σAMeasured2
√1
2ndfχ2ndf1 − 1χ2ndf2 − 1
√1
(χ2ndf1 − 1)2
+1
(χ2ndf2 − 1)2
. (6.30)
Using Equations 6.29 and 6.30, a table similar to Table 6.10 can be constructed. Table 6.11
presents the results found with this method.
To be consistent with the two methods, the systematic uncertainty estimate for Run I
is set to 10 ppb, while for Run II it is 15 ppb. The data in Run III is still being analyzed,
but preliminary findings indicate that the results of Table 6.10 and Table 6.11 are an
135
overestimate of the higher-order beam correction systematic uncertainty for the Møller
detector [72].
6.7.3 Dipole Asymmetry
A transversely polarized electron beam scattering off unpolarized electrons produces an
asymmetry with azimuthal dependence at the Møller detector. The effect is purely electro-
magnetic, due to a two photon exchange box diagram [77]. Data taken at 46 GeV, where
the beam was measured to be 85% transversely polarized in the horizontal direction, clearly
show the effect. Figure 6.17 plots the asymmetry measured in the Mid ring versus channel
number for the transverse polarization data.
Figure 6.17: Run I asymmetry plotted versus channel number for 46 GeV running.
Production data is taken with a longitudinally polarized beam, though residual trans-
verse polarization at the level of a few percent is possible. Figure 6.18 depicts the asymmetry
versus azimuth taken from the production data. Comparing the amplitudes of Figures 6.17
and 6.18 indicates that the beam is ≈ 5% transversly polarized during normal production
running, consistent with reasonable expectations.
The effect is purely electromagnetic, and so averages exactly to zero over the full az-
imuth. However, the Møller channels are statistically weighted, spoiling the azimuthal
symmetry of the detector. The dipole pattern of Figure 6.18 can then contaminate the
136
Figure 6.18: Run I asymmetry plotted versus channel number for production data.
Møller asymmetry result. Because the signal flux distribution is well centered on the face
of the Møller detector, the degree to which the dipole pattern shifts the asymmetry result
of each ring of the detector δ can be quantified by
δX = AX
N∑i=1
wi sin(2πiN
) (6.31)
and
δY = AY
N∑i=1
wi cos(2πiN
), (6.32)
where A refers to either the X or Y amplitude of the oscillation in Figure 6.18, the weights
w are defined by Equation 6.2, and the sum is taken over the N channels in the ring.
The oscillation in Figure 6.18 is dominated by the Y-dipole, due to the precession of the
longitudinal polarization in the horizontal bends of the A-Line. However, the A-Line also
has several vertical bends, incorporating a slight X-dipole oscillation into the observed
distribution. Table 6.12 gives the shift to the asymmetry measured with the Møller detector
137
Dipole Shifts (ppb)45 GeV 48 GeV
Run I -11.7 ± 3.4 -2.9 ± 1.9Run II -11.3 ± 3.4 0.4 ± 2.3
Table 6.12: The computed shifts in AMeasured due to the dipole amplitude from transversebeam polarization.
due to the dipole oscillation, computed for both Run I and Run II.
6.7.4 Spot Size Sensitivity
The detector sensitivity to beam spot size is quantified by measuring the correlation with
the wire array, as described in Section 3.4. For both Run I and Run II, the systematic
shift to AMeasured due to the spot size asymmetry was found to be consistent with zero.
A systematic uncertainty of 1 ppb is assigned to the possible systematic effect, with no
correction made.
6.7.5 Pion Background
The pion detector (Section 4.4) is used to estimate the asymmetry and dilution factor due
to pions interacting in the Møller detector. The Run I result was found to be contaminated
by particles passing through the eP detector portion of the E158 calorimeter. In Run II, the
insertable eP collimator blocked this background, making the pion detector result viable.
The Run II result will be used for both Run I and Run II. The measured dilution factor f
is 0.0014 ± 0.0011 ppb, with an asymmetry correction fA of -0.5 ± 0.8 ppb.
6.7.6 Synchrotron Radiation
The synchrotron light monitor (SLM) is used to determine the vertical polarization of the
electron beam (Section 4.5). The vertical polarization, in turn, induces a helicity-dependent
138
Diluted Synchrotron Asymmetry (ppb)45 GeV 48 GeV
Run I -1.6 ± 1.2 1.5 ± 1.2Run II -1.6 ± 1.2 2.8 ± 2.2
Table 6.13: Synchrotron correction calculated with the SLM.
asymmetry in the synchrotron radiation produced in the E158 spectrometer chicane. The
radiation produces a dilution factor f of 0.0015 ± 0.0005 for both Run I and Run II. The
radiation also represents an asymmetry correction, shown in Table 6.13.
An alternative analysis computes the vertical polarization of the beam through observing
the amplitude of the X-dipole oscillation in the plot of asymmetry versus azimuth in the
Møller detector. The oscillation is due to the two photon exchange box diagram, discussed
in Section 6.7.3. The method has the advantage that it computes the vertical polarization at
the target, whereas the SLM is located in the middle of the A-Line bend. This analysis finds
that the asymmetry contribution fA in Run I is 0.0 ± 4.5 ppb, and 0.0 ± 2.3 ppb in Run
II. The results are consistent with the SLM analysis, and they are used in the calculation
of APV .
6.7.7 Neutral Backgrounds
Neutral hadrons produced in the E158 calorimeter can penetrate the lead shielding of the
PMTs in the detector and produce a signal by directly impacting the photocathodes. Con-
tamination from hadrons produced in the eP detector that make it into the Møller PMTs
must be quantified, due to their large asymmetry.
Data was taken with PMTs blinded with aluminum tape, so that they were only sen-
sitive to neutral backgrounds. By comparing data with the quadrupole magnets off and
with the insertable eP collimator in place or removed, it was determined that the neu-
tral hadron background from the eP detector represents a dilution factor of 0.003 ± 0.001
139
for the Møller detector. The asymmetry correction is just the dilution multiplied by the
asymmetry measured in the eP detector, yielding -5 ppb ± 3 ppb, with the uncertainty
set conservatively [78]. The background is only important for Run I, because it is greatly
suppressed in Run II by the eP collimator.
A second neutral background of concern is high-energy photons that bounce from spec-
trometer collimators into the Møller detector. Data taken with the spectrometer quadrupole
magnets off determine that this background represents a dilution factor of 0.004 ± 0.002 [79].
The quadrupoles-off runs also found a large asymmetry in the Møller detector of 2.5
ppm. The most likely cause is electron-Aluminum scattering in the beam pipe. Simulations
indicate that the effect is reduced by a factor of three when the quadrupole magnets are
on. Assuming a worst-case scenario, where the entire dilution factor of 0.004 is due to this
type of background, the asymmetry correction is 2.5 ppm × 0.3 × 0.004 = 3 ppb. Due to
the large uncertainty in this analysis, a 100% uncertainty is applied to the correction.
6.7.8 Scale Factors
The beam polarization directly scales AMeasured into APV . The polarization was determined
with the polarimeter (Section 4.2.4) to be 84.9% ± 4.6% in Run I and 84.4% ± 4.6% in
Run II.
Section 5.10.1 demonstrates that the detector linearity will also directly scale the mea-
sured value of APV . The linearity of the Møller detector is found by weighting the linearity
of the In and Mid rings (Section 4.1.4) by their statistical power, yielding an overall linearity
of 99.0% ± 1.1%, in both runs.
140
Run IDilutions and Corrections
45 GeV 48 GeVfA σfA f σf fA σfA f σf
Beam 1st Order 0.0 1.2 0.0000 0.0000 0.0 1.2 0.0000 0.0000Beam 2nd Order 0.0 10.0 0.0000 0.0000 0.0 10.0 0.0000 0.0000
Dipole Bias -11.7 3.4 0.0000 0.0000 -2.9 1.9 0.0000 0.0000eP Correction -32.7 5.0 0.0795 0.0087 -34.5 5.4 0.0707 0.0077
Spot size 0.0 1.0 0.0000 0.0000 0.0 1.0 0.0000 0.0000Synchrotron 0.0 4.5 0.0015 0.0005 0.0 4.5 0.0015 0.0005
High-Energy Photons 3.0 3.0 0.0040 0.0020 3.0 3.0 0.0040 0.0020Neutral eP Leakage -5.0 3.0 0.0030 0.0010 -5.0 3.0 0.0030 0.0010
Pions -0.5 0.8 0.0014 0.0011 -0.5 0.8 0.0014 0.0011Total -46.9 13.3 0.0894 0.0090 -39.9 13.2 0.0806 0.0081
Scale FactorsBeam Polarization 0.849 ± 0.046Detector Linearity 0.990 ± 0.011
Table 6.14: The dilution factors f are dimensionless, while the asymmetry corrections fAare given in units of ppb.
6.8 Determination of APV
The results presented in Sections 6.7.1 to 6.7.8 are used with Equation 6.15 to obtain APV
from AMeasured. Tables 6.14 and 6.15 contain all of the dilution factors and asymmetry
corrections applied in Run I and Run II respectively. Table 6.16 presents the experimen-
tally measured value of APV at a Q2 of 0.026 (GeV/c)2, the primary result of the E158
experiment. The final chapter converts APV into sin2θW for comparison to theory, and uses
it to set limits on new physics phenomena.
6.9 Luminosity Monitor Results
The data from the luminosity monitor (Chapter 5) is analyzed in much the same way as the
Møller detector [81]. The expected asymmetry in the lumi signal flux is -15 ppb ± 5 ppb, an
order of magnitude smaller than APV . The detector serves as a sensitive null-asymmetry
cross-check for the results of the Møller detector. The data presented in this section is
141
Run IIDilutions and Corrections
45 GeV 48 GeVfA σfA f σf fA σfA f σf
Beam 1st Order 0.0 3.0 0.0000 0.0000 0.0 2.2 0.0000 0.0000Beam 2nd Order 0.0 15.0 0.0000 0.0000 0.0 15.0 0.0000 0.0000
Dipole Bias -11.3 3.4 0.0000 0.0000 0.4 2.3 0.0000 0.0000eP Correction -27.7 4.1 0.0648 0.0078 -29.7 4.1 0.0707 0.0077
Spot size 0.0 1.0 0.0000 0.0000 0.0 1.0 0.0000 0.0000Synchrotron 0.0 2.3 0.0015 0.0005 0.0 2.3 0.0015 0.0005
High-Energy Photons 3.0 3.0 0.0040 0.0020 3.0 3.0 0.0040 0.0020Neutral eP Leakage 0.0 0.0 0.0000 0.0000 0.0 0.0 0.0030 0.0010
Pions -0.5 0.8 0.0014 0.0011 -0.5 0.8 0.0014 0.0011Total -36.5 16.7 0.0717 0.0082 -26.8 16.5 0.0691 0.0067
Scale FactorsBeam Polarization 0.844 ± 0.046Detector Linearity 0.990 ± 0.011
Table 6.15: The dilution factors f are dimensionless, while the asymmetry corrections fAare given in units of ppb.
APV (ppb)Run I 45 GeV -183 ± 38(stat) ± 20(sys)Run I 48 GeV -158 ± 48(stat) ± 19(sys)
Run I Combined -173 ± 30(stat) ± 20(sys)
Run II 45 GeV -159 ± 44(stat) ± 23(sys)Run II 48 GeV -131 ± 37(stat) ± 22(sys)
Run II Combined -143 ± 28(stat) ± 23(sys)
TOTAL -158 ± 21 (stat) ± 17 (sys)
Table 6.16: The parity-violating asymmetry in Møller scattering.
142
Run 1 DetectorCoefficient Channel 00 Channel 02 Full Lumi
Q (ppm/ppm) -0.55 -0.80 -0.03E (ppm/MeV) -20.9 -95.7 -5.1X (ppm/µm) -7.9 55.3 -1.1Y (ppm/µm) -25.2 -4.1 0.4dX (ppm/µR) -389.8 1459.5 25.2dY (ppm/µR) -392.8 697.6 -12.9
Table 6.17: Comparison of the regression coefficients for two individual lumi channels andthe full detector. Channel 00 is at the top of the lumi ring, while channel 02 is on the rightside.
for the front lumi ring only, for simplicity. The back ring has poorer resolution, and is
consistent with the front ring in all cases.
The lumi channels are identical ion chambers, which are automatically gain matched.
The sensitivity of the detector to beam parameters is greatly suppressed when the channels
are added with equal weight, in contrast to the Møller detector in Equation 6.2. The lumi
asymmetry ARaw is the average of the individual channel asymmetries Ai, given by
ARaw =18
8∑i=1
Ai. (6.33)
The sensitivity suppression is clear in Table 6.17, which compares the regression coefficients
for two individual lumi chambers with the coefficients for the whole detector.
Table 6.1 illustrates that the lumi has sensitivity more like the Møller detector than
the Out ring, except for the conspicuous exception of the Q coefficient. The inflated Q
coefficient again indicates that not all beam parameters are being completely regressed for
the lumi. Figures 6.19 and 6.20 plot the lumi asymmetry data for Run I and Run II. The
χ2/ndf is elevated in Run I and very large in Run II, similar to the behavior of the Out
ring.
Because the lumi asymmetry is expected to be essentially zero, the mean of the system-
143
Figure 6.19: Run I lumi asymmetry data.
Figure 6.20: Run II lumi asymmetry data.
atic reversal plots, with the sign flips suppressed, is indicative of the size of the underlying
systematic effect (Figure 6.21). The Run II reversal plot indicates that the unregressed
systematic effect is at the level of 70 ppb. The effect is much smaller in Run I, most likely
due to the higher frequency of systematic reversals in the Run I data set.
6.9.1 Lumi Beam Correction Systematic Uncertainty
It is clear from the systematic reversal plots that higher-order beam correction systematic
uncertainties are potentially important for the luminosity monitor, though systematic re-
versals appear to have provided a large suppression factor. Unfortunately, because the lumi
and Out ring both have large Q coefficients, the systematic uncertainty estimation methods
144
Figure 6.21: Run I and Run II lumi asymmetry with systematic reversal sign flips ignored.
outlined for the Møller detector in Section 6.7.2.2 do not have enough statistical power to
set useful limits.
The systematic uncertainty estimate for the beam corrections to the lumi is found
through a less direct manner. First, “composite detectors” are defined by combining weight-
ing schemes of various detectors Ai to produce a composite detector Cn that is sensitive to
beam parameter n, while insensitive to all others, through
Cn =N∑i=1
wiAi, (6.34)
where the sum is taken over the N detectors used, and the weights wi are chosen to en-
hance the sensitivity of the composite monitor to a single beam parameter [80]. Table 6.18
presents a simple example of the enhanced sensitivities of a composite monitor4: the Out-
Mid combination has an enhanced sensitivity to energy fluctuations compared to the Out
or Mid detector alone.
More complicated composite monitors can be constructed to precisely isolate the sen-4The angle parameters have been converted from nRadians to nm to make their relative size with the
other beam parameters more clear.
145
Coefficient ComparisonParameter Mid Out Out - Mid
E (ppm/MeV) -26.6 76.2 102.8X (ppm/µm) 0.5 6.8 6.3Y (ppm/µm) -0.1 2.0 2.1dX (ppm/µm) -1.7 3.2 4.9dY (ppm/µm) 0.6 1.6 2.2
Table 6.18: The composite monitor Out-Mid has enhanced energy sensitivity.
Composite Monitor Coefficientsppb/keV ppb/nm
Composite E X Y dX dYCE 102.3 -0.1 -0.1 0.0 0.0CX -0.1 -22.0 0.0 -0.6 0.4CY 0.0 -0.1 -22.6 0.0 0.0CdX -0.1 0.0 0.6 24.1 0.0CdY -0.1 -0.1 0.7 0.0 -34.2
Table 6.19: Regression coefficients of the composite monitors Cn.
sitivity to one beam parameter. The composite monitors are made up of combinations of
the front lumi, the In, Mid, and Out rings of the Møller detector, and the X and Y dipole
weighting scheme for each detector. It is expected that the unregressed systematic effects
will be greatly enhanced in the composite monitors.
Table 6.19 details the composite sensitivities achieved with this method. The composite
monitors Cn are each composed of roughly seven of the original detectors. The In, Mid,
and Out ring are always added in a way to ensure that the physics asymmetry APV cancels
between them, leaving only the difference in the eP corrections.
The asymmetry measured with each composite monitor is calculated and compared to
the expected value. The expectation can be non-zero, due to differences in the eP corrections
of the Møller detector regions, and the vertical polarization of the beam can induce real
dipole asymmetries. The systematic uncertainty δn assigned to the regression correction of
146
Run I Front Lumi Systematic Uncertainty EstimatesParameter ∆ cComp clumi suppression α systematic error (ppb)
E 92.3 ± 50.1 102.3 -5.1 -0.050 -4.6 ± 2.5X -55.8 ± 26.5 -22.0 -1.1 0.050 -2.8 ± 1.3Y -139.2 ± 53.9 -22.6 0.4 -0.018 2.5 ± 1.0dX 18.2 ± 60.2 24.1 0.6 0.025 0.5 ± 1.5dY 117.5 ± 70.9 -34.2 -0.3 0.009 1.0 ± 0.6
TOTAL: -3.4 ± 6.9
Table 6.20: Run I systematic uncertainty estimates for the front lumi. The columns labeledcComp and clumi contain the dominant regression coefficient for the composite monitor andthe lumi, respectively.
Run II Front Lumi Systematic Uncertainty EstimatesParameter ∆ cComp clumi suppression systematic error (ppb)
E 120.8 ± 44.6 67.4 -5.8 -0.086 -10.4 ± 3.8X 63.8 ± 21.0 -14.6 -1.4 0.096 6.1 ± 2.0Y -46.7 ± 41.6 -22.4 -0.8 0.036 -1.7 ± 1.5dX 123.0 ± 60.2 25.2 2.9 0.115 14.2 ± 6.9dY 135.2 ± 75.8 -33.4 0.6 -0.018 -2.4 ± 1.4
TOTAL: 5.8 ± 15.6
Table 6.21: Run II systematic uncertainty estimates for the front lumi. The columns labeledcComp and clumi contain the dominant regression coefficient for the composite monitor andthe lumi, respectively.
the nth parameter for the lumi is then
δn = αn(CMeasuredn − CExpectedn ) → αn∆n. (6.35)
The coefficient αn is the ratio of the the nth lumi regression coefficient to the sensitive
coefficient of the composite monitor Cn. The ratio expresses the relative sensitivity of the
lumi monitor to the composite monitor for beam parameter n. Table 6.20 and Table 6.21
relate the systematic uncertainties calculated for the lumi in this manner for Run I and
Run II. The total systematic uncertainty is found by combining the individual systematic
uncertainties and errors linearly, to be conservative.
The uncertainty assigned for beam corrections for the front lumi is 7 ppb in Run I and
147
16 ppb in Run II. The uncertainty is still quite small compared to the variations observed
in Figure 6.21. It is clear that the energy and halfwave plate systematic reversals provided
large cancellation of the unregressed systematic effect and were critical for the lumi result.
6.9.2 Lumi Dipole Contamination
Unlike the Møller detector, the lumi channels are not weighted in forming the asymmetry of
the whole detector, so the dipole contamination shifts in Equation 6.31 and Equation 6.32
from the Møller detector analysis in Section 6.7.3 are identically zero. However, since the
lumi is not necessarily aligned with the center of the Møller detector, and hence the beam
path, a large position offset at the lumi could still induce a dipole shift. Figure 6.22 displays
the individual channel asymmetries for the lumi in Run I. There is no indication of any large
offset, since the sinusoid fits well. Therefore, no systematic uncertainty will be assigned to
the lumi for this effect.
Figure 6.22: Run I lumi asymmetry plotted versus channel number.
148
6.9.3 Lumi Dilutions and Scale Factors
The measured lumi asymmetry must be corrected for backgrounds and uncertainties through
Equation 6.15. The only background for the lumi comes from synchrotron radiation. In
Section 5.9, the synchrotron dilution factor f , averaging over the whole detector, is found
to be 0.0035 ± 0.0004. The uncertainty is set by the target-out signal seen in the vertical
chambers.
The asymmetry correction for the synchrotron background would require a detailed
simulation of the lumi to determine its analyzing power for vertical polarization. This
analysis is underway, but is not yet completed. However, the result is expected to be very
similar to the Møller detector result, so the Møller detector value is taken as an estimate,
scaled by the ratio of the dilution factors for the two detectors. In Run I, the synchrotron
asymmetry correction is 0.0 ± 10.5 ppb, and in Run II it is 0.0 ± 5.4 ppb.
The only additional asymmetry correction is due to sensitivity to the beam spot size.
The spot size systematic was found to be ±1 ppb for Run I and ±2 ppb for Run II in
Section 3.4.
The lumi has the same two scale factors as the Møller detector: beam polarization and
detector linearity. The linearity used is 99 ± 2%, based on the results of Section 5.10.
6.10 Lumi Asymmetry Result
Combining the corrections and systematic uncertainties of Sections 6.9.1 and 6.9.3 with
the measured asymmetries in Section 6.9 yields the final luminosity monitor asymmetry.
Table 6.22 presents the lumi results. Both the Run I and Run II data are consistent with
the predicted value of -15 ppb ± 5 ppb. The result is a consistency cross-check for the
149
Luminosity Monitor Asymmetry ResultsRun I -19.6 ppb ± 17.6 ppb (stat) ± 15.2 ppb (sys)Run II -16.5 ppb ± 14.1 ppb (stat) ± 20.4 ppb (sys)Total -18.1 ppb ± 11.0 ppb (stat) ± 17.6 ppb (sys)
Table 6.22: Luminosity monitor asymmetry results.
asymmetry measured with the Møller detector.
150
Chapter 7
The Weak Mixing Angle
This chapter presents the result for sin2θW calculated from the SLAC E158 measurement
of the parity-violating asymmetry APV in Møller scattering. The result is then used to set
limits on possible extensions to the Standard Model. Finally, future experiments planned
to measure the running of sin2θW are discussed.
7.1 Extraction of sin2θW
The QED calculation of APV is covered in Chapter 2. Using the convention for sin2θeffW
given in Equation 2.14, the asymmetry is given by
APV =−GµQ2
√2πα
1 − y
1 + y4 + (1 − y)4Fb(y)(1 − 4 sin2 θeffW ), (7.1)
where y is related to the center-of momentum scattering angle θCM through
y =1 − cos θCM
2. (7.2)
The scale factor Fb(y) is included to account for bremsstrahlung effects that depend
on the geometry of the experiment [82]. The relevant diagrams are depicted in Figure 7.1.
151
Bremsstrahlung radiation before scattering effectively lowers the beam energy, while radiat-
ing afterward affects the experimental acceptance. The overall effect of Fb was determined
to be small, with an average value of 1.01 ± 0.01.
Figure 7.1: Bremsstrahlung diagrams included in Fb(y). The crossed versions must also becomputed, for a total of 16 diagrams.
The analyzing power AP is defined as the flux weighted average over the kinematic
factors in Equation 7.1 so that
APV = AP(1 − 4 sin2 θeffW ). (7.3)
The analyzing power is computed using the same Monte-Carlo simulation that was employed
for background subtraction in Section 6.7.1. Table 7.1 presents the computed values for
AP, where In and Mid refer to particular regions of the Møller detector (Section 4.1). The
uncertainty on AP was found by varying the parameters of the simulation within reasonable
bounds. The simulation was also used to determine that the average Q2 is 0.026 (GeV/c)2.
152
Analyzing Power (ppm)Run I 45 GeV 48 GeV
In -2.976 ± 0.060 -3.171 ± 0.060Mid -3.304 ± 0.053 -3.539 ± 0.042
Run II 45 GeV 48 GeVIn -3.046 ± 0.061 -3.182 ± 0.060
Mid -3.372 ± 0.044 -3.537 ± 0.042Average: -3.298 ± 0.051
Table 7.1: Analyzing powers computed from simulation. The overall average AP is deter-mined by weighting the entries with the corresponding uncertainty on APV .
Combining the average analyzing power from Table 7.1,
−3.298 ppm ± 0.051 ppm, (7.4)
and the measured value of APV from Table 6.16,
−158 ppb ± 21 ppb (stat) ± 17 ppb (sys), (7.5)
the weak mixing angle is found to be
sin2 θeffW = 0.2380 ± 0.0016 (stat) ± 0.0013 (sys). (7.6)
For comparison, the theoretical prediction is
sin2 θeffW = 0.2385 ± 0.0006 (theory). (7.7)
It is clear that the agreement between the experimentally measured value and the theoretical
153
prediction is quite good. The difference between them is
∆ sin2 θeffW = −0.0005 ± 0.0021, (7.8)
corresponding to ≈ 0.25 standard deviations.
Equivalently, using the convention for sin2θW (Q2) preferred by theorists (Equation 2.13),
the E158 result is
sin2 θMSW = 0.2376 ± 0.0020. (7.9)
Figure 7.2 presents the E158 value for sin2θMSW on a plot along with the theoretically pre-
dicted Q2 dependence. Figure 7.3 presents the measurements of the weak mixing angle
evolved to the Z0 mass, assuming the Standard Model, to aid in the comparison.
Figure 7.2: Experimental results and the theoretical running of the weak mixing angle.
154
Figure 7.3: The measured weak mixing angle evolved to the Z0 mass.
7.2 Physics beyond the Standard Model
The deviation of the measured value of sin2θeffW from the theoretical predication can be
used to set limits on possible extensions of the Standard Model. The following sections use
the deviation in Equation 7.8 to set the E158 constraints on new physics.
7.2.1 Electron Compositeness Limit
In the current model of particle physics, the electron is assumed to be a truly point-like
particle. However, it is possible that the electron has substructure [83]. In electron-electron
scattering, electron compositeness can be cast as a contact interaction, with the general
Lagrangian given as
L =4π2Λ2
[ηLLψLγµψLψLγµψL + ηRRψRγµψRψRγ
µψR + 2ηRLψRγµψRψLγµψL]. (7.10)
The Λ term parameterizes the energy scale at which the contact interaction becomes im-
portant. The indices L and R refer to the chiral components of the interaction, signifying
155
left or right helicity. The η coefficients are free parameters, with the constraint that the
largest member have a magnitude of unity or less. There are no symmetry requirements
that the interaction should conserve parity, so the η values are independent.
The Lagrangian produces an amplitude that is purely real, and at low Q2 there is an
interference term with the dominant electromagnetic diagrams. In contrast, at the Z0
resonance where the dominant amplitude is purely imaginary, no interference term results.
Low energy experiments are therefore inherently more sensitive to new physics in the form
of Equation 7.10 than the Z0 pole experiments.
The process described by Equation 7.10 produces a shift in the predicted value of sin2θW
given by
∆ sin2 θeffW =π√2GF
ηLL − ηRRΛ2
. (7.11)
For parity-violation experiments, it is conventional to set ηLL to ±1 and ηRR to zero to
obtain limits on Λ±LL. The E158 compositeness constraints are then
Λ+LL > 8.0 TeV/c2 (7.12)
and
Λ−LL > 6.9 TeV/c2, (7.13)
at the 95% confidence level.
The present limit on this quantity is from the combined data of the Aleph, Delphi, L3,
and Opal detectors at LEP [84]. They report a constraint on compositeness of
Λ+LL > 8.3 TeV/c2 (7.14)
156
and
Λ−LL > 10.3 TeV/c2. (7.15)
Individual results from these experiments yield constraints in the range of three to five
TeV/c2. The E158 limit is quite competitive with the collider results.
7.2.2 Scalar Doubly Charged Higgs Limit
Møller scattering is uniquely sensitive to s-channel scattering through exchange of an exotic
doubly charged scalar Higgs boson. Particles such as this often appear in theories beyond
the Standard Model and are not required to respect parity [85]. Figure 7.4 depicts the
diagram for this process.
Figure 7.4: Doubly charged Higgs particle exchange diagram.
The mass of the Higgs particle is constrained to be much larger than the interaction
energy, so it can be described as a contact interaction in the same formalism as the electron
compositeness limit. The process would shift sin2 θeffW by
∆ sin2 θeffW =√
28
(g
MH
)2 ηLL − ηRRGF
, (7.16)
where g/MH is the ratio of the coupling to the mass of the Higgs particle. Again, ηLL is
set to ±1 while ηRR = 0 to obtain the limit on(
gMH
)2
LL± . The 95% confidence limits from
157
the E158 result are (g
MH
)2
LL+< 0.017GF (7.17)
and (g
MH
)2
LL−< 0.022GF . (7.18)
The current limits are set by observing the related process of muonium to anti-muonium
conversion [86]. Figure 7.5 depicts the diagram for this process. These experiments place
Figure 7.5: Muonium to anti-muonium conversion.
the constraint (g
MH
)2
< 0.14GF , (7.19)
on the doubly charged Higgs parameters, at the 90% confidence level.
The E158 limit represents an order of magnitude improvement. However, it should be
noted that the muonium experiments would still be sensitive even in the absence of parity
violation.
7.2.3 Extra Z Boson Limit
Many extensions to the Standard Model include new heavy analogues to the Z0. The new
particle is usually denoted as a Z′. Some Grand Unified Theories embed the Standard
Model in the group E6, producing two new Z′ states [87]. The extra Z particles assume the
158
forms [88]
Zβ = Zχ cos β + Zψ sin β (7.20)
and
Zβ′ = −Zχ sin β + Zψ cos β. (7.21)
The Zχ state has distinct couplings to different fermions, while the Zψ state couples uni-
versally to all fermions [89]. The presence of these new particles scales the prediction for
APV [32] by
1 + 7
m
2Z0
m2Zβ
(cos2 β +
√53
sinβ cos β
)+m2Z0
m2Zβ′
(sin2 β −
√53
sin β cos β
) . (7.22)
The mixing angle β is model dependent. In the SO(10) model, a group of considerable
theoretical interest, β = 0 and Equation 7.22 simplifies to
1 + 7m2Z0
m2Zχ
. (7.23)
The weak mixing angle would be modified as
1 − 4 sin2 θeffW →(
1 + 7M2Z0
M2Zχ
)(1 − 4 sin2 θeffW
). (7.24)
The E158 measurement of the weak mixing angle can then be used to set the limit on the
mass of the extra Z particle of
MZχ > 410 GeV/c2, (7.25)
at the 95% confidence level. The single highest current constraint is provided by CDF which
reports Zχ > 595 GeV/c2 at the 95% confidence level [90].
159
7.2.4 Oblique Parameter X Limit
Extensions to the Standard Model could be manifested through radiative corrections to
the gauge boson propagators, known as oblique corrections. The standard Peskin-Takeuchi
parameters STU were introduced to quantify deviations from the Standard Model due to
oblique corrections for energies much greater than MZ0 [91]. This formalism was extended
to include new physics effects at the weak scale by the introduction of three new parameters
VWX [92].
The running of sin2θW from the Z0 resonance down to low Q2 can be shown to isolate
the X parameter [93]. The κ(0) term in Equation 2.8 is modified as [32]
κ(0) → (1 − 0.032X)κ(0). (7.26)
The shift in sin2 θeffW is then
∆ sin2 θeffW = −ρκ(0)(0.032X) sin2 θMSW (M2
Z0). (7.27)
The quantities ρ and κ are defined in the theoretical calculation of APV in Sections 2.4.4
and Section 2.4.1. Using the E158 result, it is found that the limit on X is
X = 0.07 ± 0.28. (7.28)
The value of X based on a global STUVWX fit to world data in 1994 has an uncertainty
of 0.58 [94]. The inclusion of the E158 result, and to an equal degree the atomic parity
violation result, will produce a substantial improvement.
160
7.3 Future Experiments
Several upcoming experiments will be conducted to map out the running of the weak mixing
angle. Figure 7.6 depicts the proposed experiments and the expected final uncertainties.
The following sections give brief descriptions of these experiments.
Figure 7.6: Present and future experiments used to map the running of the weak mixingangle.
7.3.1 E158 Run III
The final run of the E158 experiment, Run III, took place in the Summer of 2003. The
amount of data collected was approximately equal to the sum of Run I and Run II. The
addition of the Run III data is therefore expected to reduce the uncertainty on sin2θW by
a factor of ≈ √2. Additionally, the analysis of the “sliced” BPM signals (Section 6.7.2.2)
is expected to reduce the systematic uncertainty on the Run I and Run II results. Overall,
the E158 result should yield an uncertainty of approximately
δ sin2 θeffW = ±0.0014. (7.29)
161
7.3.2 QWeak
The QWeak experiment has been approved to run in Hall C at Jefferson Lab, beginning in
2007 [95]. It will operate at a Q2 of 0.03 (GeV/c)2, almost identical to the E158 exper-
iment. QWeak will employ a 1.2 GeV longitudinally-polarized electron beam on a 35 cm
liquid hydrogen target to observe the parity-violating asymmetry in elastic electron-proton
scattering.
In the limit of a small scattering angle, the asymmetry APVQWeak is given by
APVQWeak =−GF
4πα√
2
[Q2(1 − 4 sin2 θW ) +Q4B(Q2)
]≈ −300 ppb. (7.30)
The term B(Q2) is due to the electromagnetic and weak form factors of the proton, and
contributes roughly 1/3 of the total asymmetry. It will be determined from the results of
other experiments, including HAPPEX and SAMPLE.
The goal is to perform a 4% measurement of APVQWeak, which would correspond to a 0.3%
measurement of sin2θW . The dominant sources of systematic uncertainty on APVQWeak are
anticipated to be a 2% uncertainty coming from the determination of B(Q2) and a 1.5%
uncertainty from the beam polarization measurement.
7.3.3 DIS-Parity at JLab
The DIS-Parity experiment has been proposed to run in Hall A at Jefferson Lab [96]. The
experiment will use a longitudinally-polarized electron beam on a 15 cm liquid deuterium
target, to measure the parity-violating asymmetry ADIS in deep-inelastic scattering. The
Q2 will be 2.0 (GeV/c)2, placing it between the NuTeV and the E158 results. The parity-
162
violating asymmetry is given by
ADIS = (109 ppm)Q2[(−3
2+
103
sin2 θW
)+ Y Rv
(−32
+ 6 sin2 θW
)]∼ −100 ppm,
(7.31)
where Y ≈ 1 contains kinematics information and Rv ≈ 1 depends on quark distributions.
The goal of the first run of the experiment will be a determination of sin2θW to ≈ 1%
using a 6 GeV beam. If a large deviation from the Standard Model prediction is observed,
a second run is planned that would employ the 12 GeV upgraded beam at JLab and run at
a larger scattering angle [97].
7.3.4 The Large Hadron Collider
The Large Hadron Collider (LHC) at CERN is expected to become operational in 2008 [98].
It is designed to produce proton-proton collisions with an energy of 14 TeV. By observing a
large number of Z0 decays, it is expected that sin2θW (M2Z0) will be measured to a precision
of ± 0.0001. Also, the large energy scale will allow for direct searches of extra Z bosons,
superseding all indirect constraints. In particular, the Zχ in the SO(10) model will be
constrained at the level of 4.5 TeV, an order of magnitude improvement.
7.3.5 Next Linear Collider
The high-energy lepton-lepton collisions of the proposed “Next Linear Collider” (NLC)
would be required to measure the weak mixing angle for Q2 much larger than M2Z0 [99]. If
the machine is ever constructed, it is expected that collision energies in the range of 1 to
1.5 TeV will be achieved, as shown in Figure 7.6 [100].
163
7.4 Conclusion
The E158 result represents the first measurement of parity violation in Møller scattering.
The observed value of sin2 θeffW at a Q2 of 0.026 (GeV/c)2 is in agreement with the Standard
Model prediction for the running of the weak mixing angle. The success of the E158 experi-
ment demonstrates the technical feasibility of measuring small parity-violating asymmetries,
setting the stage for future challenging parity violation experiments that will further probe
the Standard Model of particle physics.
164
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